Totally Positive Matrices (Cambridge Tracts in Mathematics)

C AMB RI D G E TR ACTS I N M AT H E M AT I C S General Editors ´ S , W . FU LTO N , A . K AT O K , F. K I RWA N , B . B...

Author: Pinkus Allan

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C AMB RI D G E TR ACTS I N M AT H E M AT I C S General Editors ´ S , W . FU LTO N , A . K AT O K , F. K I RWA N , B . B O LLO B A P. S A RN A K, B . S I M O N , B . T O TA RO 181 Totally Positive Matrices

Totally Positive Matrices ALLAN PINKUS Technion, Israel

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ ao Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521194082 c A. Pinkus 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-19408-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

This monograph is dedicated to the memory of I. J. Schoenberg, M. G. Krein, F. R. Gantmacher and S. Karlin, the four pioneers of the theory of total positivity. We work in the dark – we do what we can – we give what we have. Our doubt is our passion, and our passion is our task.

Contents

Foreword 1

page ix

Basic properties of totally positive and strictly totally positive matrices 1.1 Preliminaries 1.2 Building (strictly) totally positive matrices 1.3 Nonsingularity and rank 1.4 Determinantal inequalities 1.5 Remarks

1 1 5 12 24 33

Criteria for total positivity and strict total positivity 2.1 Criteria for strict total positivity 2.2 Density and some further applications 2.3 Triangular total positivity 2.4 LDU factorizations 2.5 Criteria for total positivity 2.6 “Simple” criteria for strict total positivity 2.7 Remarks

36 37 41 47 50 55 60 74

3

Variation diminishing 3.1 Main equivalence theorems 3.2 Intervals of strict total positivity 3.3 Remarks

76 76 83 85

4

Examples 4.1 Totally positive kernels and Descartes systems 4.2 Exponentials and powers 4.3 Cauchy matrix 4.4 Green’s matrices 4.5 Jacobi matrices

87 87 88 92 94 97

2

vii

viii

Contents 4.6 4.7 4.8 4.9 4.10 4.11

5

Hankel matrices Toeplitz matrices Generalized Hurwitz matrices More on Toeplitz matrices Hadamard products of totally positive matrices Remarks

101 104 111 117 119 125

Eigenvalues and eigenvectors 5.1 Oscillation matrices 5.2 The Gantmacher–Krein theorem 5.3 Eigenvalues of principal submatrices 5.4 Eigenvectors 5.5 Eigenvalues as functions of matrix elements 5.6 Remarks

127 127 130 140 144 149 152

Factorizations of totally positive matrices 6.1 Preliminaries 6.2 Factorizations of strictly totally positive matrices 6.3 Factorizations of totally positive matrices 6.4 Remarks Afterword References Author index Subject index

154 154 156 164 167 169 174 180 182

6

Foreword

In this monograph was present the central properties of finite totally positive matrices. As such, the monograph has only six main chapters. We consider the basic properties of such matrices, determinantal criteria for when a matrix is totally positive, their variation diminishing properties, various examples of totally positive matrices, their eigenvalue/eigenvector properties, and factorizations of such matrices. Numerous topics are excluded from this exposition. Total positivity is a theory of considerable consequence, and the most glaring omissions of this monograph are undoubtedly its various applications to diverse areas. Aside from the many applications mentioned in Gantmacher, Krein [1950] and Karlin [1968], applications can be found in approximation theory (see Schumaker [1981], Pinkus [1985c]), combinatorics (see Brenti [1989], [1995], [1996]), graph theory (see Fomin, Zelevinsky [2000], Berenstein, Fomin, Zelevinsky [1996]), Lie group theory (see Lusztig [1994]), majorization (see Marshall, Olkin [1979]), noncommutative harmonic analysis (see Gross, Richards [1995]), shape preservation (see Goodman [1995]), computing using totally positive matrices (see de Boor, Pinkus [1977], Koev [2005], Demmel, Koev [2005], Koev [2007]), refinement equations and subdivision (see Cavaretta, Dahmen, Micchelli [1991], Micchelli, Pinkus [1991]), and infinite totally positive banded matrices (see Cavaretta, Dahmen, Micchelli, Smith [1981], de Boor [1982], Smith [1983], Dahmen, Micchelli, [1986]). See also the many references in these papers and also the many references to these papers. There has been no attempt to make this monograph all-encompassing, and we apologize to all who feel that their contributions to the theory have been slighted as a consequence. The theory of totally positive matrices is an odd bird in the matrix theory aviary. Much of the motivation for its study has come from problems

ix

x

Foreword

in analysis, and the main initiators and contributors to the theory were analysts. I. J. Schoenberg was interested in the problem of estimating the number of real zeros of a polynomial, and this led him to his work on variation diminishing transformations (in the early 1930s) and P´ olya frequency sequences, functions, and kernels (late 1940s and early 1950s). These, together with his work on splines (1960s and 1970s), are central topics in the theory of total positivity. M. G. Krein was led to the theory of total positivity via ordinary differential equations whose Green’s functions are totally positive (mid 1930s). S. Karlin came to the theory of total positivity (in the 1950s and 1960s) by way of statistics, reliability theory, and mathematical economics. The two major texts on the subject Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, by F. R. Gantmacher and M. G. Krein (see Gantmacher, Krein [1950]), and Total Positivity. Volume 1, by S. Karlin (see Karlin [1968]), are a blend of analysis and matrix theory (and in the latter case the emphasis is most certainly on analysis). (Their companion volumes The Markov Moment Problem and Extremal Problems, by M. G. Krein and A. A. Nudel’man (see Krein, Nudel’man [1977]) and Tchebycheff Systems: with Applications in Analysis and Statistics, by S. Karlin and W. J. Studden (see Karlin, Studden [1966]), are totally devoted to topics of analysis.) Thankfully we have the short monograph of T. Ando that eventually appeared as Ando [1987] (it was written a few years earlier) and was devoted to totally positive matrices. The present monograph is an attempt to update and expand upon Ando’s monograph. A considerable amount of research has been devoted to this area in the past twenty years, and such an update is certainly warranted. It was Schoenberg, in Schoenberg [1930], who coined the term total positiv (in German). Krein and Gantmacher (see Gantmacher, Krein [1935]), unaware of Schoenberg’s earlier paper, used the term complètement non négative and complètement positive (French) for totally positive and strictly totally positive, respectively. As such, many authors use the term totally nonnegative and totally positive for totally positive and strictly totally positive, respectively, which, aside from the lack of consistency and order, all too often leads to confusion. We follow the Schoenberg/Karlin/Ando terminology. It is a pleasure to acknowledge the help of Carl de Boor and David Tranah. All errors, omissions and other transgressions are the author’s responsibility.

Foreword

xi

I would like to close this short foreword with a personal note. My first mathematical paper (jointly written with my doctoral supervisor Sam Karlin) was in the area of total positivity. It is said that as one gets old(er) one often returns to one’s first love. I plead guilty on both counts. Haifa, 2008.

1 Basic properties of totally positive and strictly totally positive matrices

In this chapter, we introduce some of the notation, basic definitions, and various classic facts and formulæ. Many of the results of this chapter will be used in subsequent chapters. Matrix notation, more especially the notation used for submatrices and minors, is both clumsy and problematic. It definitely takes getting used to. But rest assured that one does eventually get used to it. The medium here is not the message. In Section 1.2 we consider some simple and less simple operations that preserve total positivity and strict total positivity. Section 1.3 is about nonsingularity and rank. Here we immediately see results that are less than obvious. Finally, in Section 1.4, we present a few basic determinantal inequalities that are valid for totally positive and strictly totally positive matrices.

1.1 Preliminaries For a positive integer n, and for each p ∈ {1, . . . , n}, we define the simplex Ipn := {i = (i1 , . . . , ip ) : 1 ≤ i1 < · · · < ip ≤ n} in Zp+ . That is, Ipn denotes the set of strictly increasing sequences of p integers in {1, . . . , n}. We use the following notation to define submatrices and minors of a n matrix. If A = (aij )ni=1 m j=1 is an n × m matrix, then for each i ∈ Ip and m j ∈ Iq we let i1 , . . . , ip i p =A := (aik j )k=1 q=1 A[i, j] = A j j1 , . . . , jq denote the p × q submatrix of A determined by the rows indexed i1 , . . . , ip 1

2

Basic properties of totally positive matrices

and columns indexed j1 , . . . , jq . When p = q then i1 , . . . , ip i p =A := det (aik j )k,=1 A(i, j) = A j j1 , . . . , jp denotes the associated minor, i.e., the determinant of the submatrix. This monograph is about totally positive and strictly totally positive matrices. They are defined as follows: Definition 1.1 An n × m matrix A is said to be totally positive (TP) if all its minors are nonnegative, i.e., i1 , . . . , ip ≥0 (1.1) A(i, j) = A j1 , . . . , jp for all i ∈ Ipn , j ∈ Ipm , and all p = 1, . . . , min{n, m}. It is said to be strictly totally positive (STP) if strict inequality always holds in (1.1). We use and reuse various classic facts and formulæ. We list some of them here for easy reference. Cauchy–Binet and p th compound matrices The p th compound mof the n × m matrix A is denoted by A[p] and is defined as the matrix n p × p matrix with entries (A(i, j))i∈Ipn ,j∈Ipm where the i ∈ Ipn and j ∈ Ipm are arranged in lexicographic order, i.e., for distinct i, k ∈ Ipn we set i > k if the first nonzero term in the sequence i1 − k1 , . . . , ip − kp is positive. Assume B = CD, where B is an n × m matrix, C is an n × r matrix, and D is an r × m matrix. The Cauchy–Binet formula may be written as follows. For each i ∈ Ipn , j ∈ Ipm , C(i, k)D(k, j) , B(i, j) = k∈Ipr

i.e.,

B

i1 , . . . , ip j1 , . . . , jp

=

1≤k1 0 we have ars = 0. Similarly, if > j then it follows that the left shadow of aij is zero. Let us now assume that r > 1. As i + 1, . . . , i + r A j + 1, . . . , j + r is of rank r − 1 there are p, q ∈ {1, . . . , r} such that

i + 1, . . . , i + p, . . . , i + r > 0. A j + 1, . . . , j + q, . . . , j + r Set

bk = A

i + 1, . . . , i + p, . . . , i + r, k j + 1, . . . , j + q, . . . , j + r,

(1.3)

for k ∈ {1, . . . , n}\{i + 1, . . . , i + p, . . . , i + r} and ∈ {1, . . . , m}\{j + 1, . . . , j + p, . . . , j + r}, where we understand that the row and column indices are rearranged in increasing order. Let B = (bk ). By Sylvester’s Determinant Identity, we see that B is an (n − r + 1) × (m − r + 1) totally positive matrix and bi+p,j+q = 0. As such we can apply the result proved in the case r = 1.

1.3 Nonsingularity and rank

15

+ q, . . . , j + r}, i.e., If bi+p, = 0 for all ∈ {1, . . . , m}\{j + 1, . . . , j i + 1, . . . , i + r A =0 j + 1, . . . , j + q, . . . , j + r, for all , then from (1.3) it follows that the rows i + 1, . . . , i + r are linearly dependent. Similarly, if bk,j+q = 0 for all k ∈ {1, . . . , n}\{i + 1, . . . , i + p, . . . , i + r}, then the columns j + 1, . . . , j + r are linearly dependent. If the right shadow of bi+p,j+q vanishes, i.e.,

i + 1, . . . , i + p, . . . , i + r, k A =0 j + 1, . . . , j + q, . . . , j + r, for all k ≤ i + p, ≥ j + q, then from (1.3) we see that 1, . . . , i + r A j + 1, . . . , m is of rank r − 1. Similarly, if the left shadow of bi+p,j+q vanishes, then i + 1, . . . , n A 1, . . . , j + r is of rank r − 1. Proposition 1.15, in the case r = 1, is sometimes referred to as the “shadow lemma.” We now prove Theorem 1.13. Proof of Theorem 1.13 We first prove directly that arr > 0 for all r ∈ {1, . . . , n}. Assume arr = 0. From Proposition 1.15 we have four options. But all four options contradict the nonsingularity of A. Obviously we cannot have that the rth row or column of A is zero. Thus either the left or right shadow of arr is zero. Assume it is the right shadow. Then aij = 0 for all i ≤ r and all j ≥ r, implying that the first r rows of A are linearly dependent. This is a contradiction and therefore arr > 0. We derive the general result by applying an induction argument on the size of the minor and using Sylvester’s Determinant Identity. We assume that for any totally positive nonsingular n × n matrix (any n) all principal minors of order at most p − 1 are strictly positive (p ≤ n). We prove that this same result holds for all principal minors of order p . We have proved the case p = 1. For any 1 ≤ i1 < · · · < ip ≤ n set i1 , . . . , ip−1 , k bk = A , i1 , . . . , ip−1 ,

16


for k, ∈ {1, . . . , n}\{i1 , . . . , ip−1 }, and let B = (bk ). As an immediate consequence of Sylvester’s Determinant Identity and our induction hypothesis, it follows that B is totally positive and nonsingular. Thus the diagonal entries of B are strictly positive. As p−2 i1 , . . . , ip−1 i1 , . . . , ip 0 < bip ip = A A i1 , . . . , ip−1 i1 , . . . , ip and, by our induction hypothesis, we have i1 , . . . , ip−1 >0 A i1 , . . . , ip−1 it therefore follows that

A

i1 , . . . , ip i1 , . . . , ip

> 0.

For totally positive matrices there is also an interaction between its rank, which of its rows and columns can be linearly (in)dependent, and the strict positivity of specific minors. This we detail in the next series of results. Proposition 1.16 Let A be an n × m totally positive matrix, and let ak denote the kth row of A, k = 1, . . . , n. Given 1 = i1 < · · · < ir+1 = n, assume that the r + 1 vectors ai1 , . . . , air+1 are linearly dependent, while the r vectors ai1 , . . . , air and ai2 , . . . , air+1 are each linearly independent. Then A is necessarily of rank r. Proof Since the ai1 , . . . , air+1 are linearly dependent, while the ai2 , . . . , air+1 are linearly independent we can write a i1 =

r+1

cs ais .

s=2

As the ai1 , . . . , air are linearly independent there exist 1 ≤ j1 < · · · < jr ≤ m for which i1 , . . . , ir A > 0. j1 , . . . , jr Substituting for ai1 we obtain r+1 is , i2 , . . . , ir ir+1 , i2 , . . . , ir i1 , . . . , ir = = cr+1 A cs A A j1 , . . . , jr j1 , j2 , . . . , jr j1 , j2 , . . . , jr s=2 i2 , . . . , ir , ir+1 . = (−1)r+1 cr+1 A j1 , . . . , jr−1 , jr


17

The matrix A is totally positive and the left-hand side is strictly positive. Thus (−1)r+1 cr+1 > 0. By assumption A is of rank at least r. If r = m there is nothing to prove, while if r + 1 = n there is also nothing to prove. As such we assume that r + 1 ≤ m and r + 1 < n. Let ∈ {1, . . . , n}\{i1 , . . . , ir+1 }. Thus 1 = i1 < < ir+1 = n. For every choice of 1 ≤ k1 < · · · < kr+1 ≤ m r+1 is , i2 , . . . , , . . . , ir i1 , . . . , , . . . , ir = cs A A k1 , k2 , . . . , kr+1 k1 , k2 , . . . , kr+1 s=2 ir+1 , i2 , . . . , , . . . , ir = cr+1 A k1 , k2 , . . . , kr+1 i2 , . . . , , . . . , ir , ir+1 = (−1)r cr+1 A . k1 , k2 , . . . , kr+1 As A is totally positive and (−1)r cr+1 < 0, it follows that i1 , . . . , , . . . , ir A = 0. k1 , k2 , . . . , kr+1 Since this is true for every choice of 1 ≤ k1 < · · · < kr+1 ≤ m, we have that a ∈ span{ai1 , . . . , air }. From our assumption we also have air+1 ∈ span{ai1 , . . . , air }. Thus A is of rank r. Proposition 1.17 Let A be an n × m totally positive matrix. Assume 1 = i1 < · · · < ir+1 = n and 1 = j1 < · · · < jr+1 = m. If i1 , . . . , ir+1 = 0, A j1 , . . . , jr+1 while

A

i1 , . . . , ir j1 , . . . , jr

,A

i2 , . . . , ir+1 j2 , . . . , jr+1

> 0,

then A is of rank r. Proof Let ak denote the kth row of A. By assumption the r vectors ai1 , . . . , air and the r vectors ai2 , . . . , air+1 are linearly independent. Thus it suffices, by Proposition 1.16, to prove that the r + 1 vectors ai1 , . . . , air+1 are linearly dependent. This latter result follows from an application of Proposition 1.16 to the (r + 1) × m matrix i1 , . . . , ir+1 B=A 1, . . . , m

18


where we exchange the roles of the rows and columns. That is, if bk denotes the kth column of B, then, by assumption, the r + 1 vectors bj1 , . . . , bjr+1 are linearly dependent, while the r vectors bj1 , . . . , bjr and bj2 , . . . , bjr+1 are each linearly independent. Thus B is of rank r, implying that the r + 1 vectors ai1 , . . . , air+1 are linearly dependent. Another consequence of Proposition 1.16 is the following, which tells us something about the possible singularity structure of totally positive matrices. Proposition 1.18 Let A be an n × m totally positive matrix, and let ak denote the kth row of A, k = 1, . . . , n. Given 1 = i1 < · · · < ir+1 < n, assume that the r + 1 vectors ai1 , . . . , air+1 are linearly independent. Let k > ir+1 (k ≤ n) and assume that the r + 1 vectors ai1 , . . . , air , ak are linearly dependent. Then ak = 0. Proof This proposition can be proved using the method of proof in Proposition 1.16. But it also follows as a consequence of Proposition 1.16. We explain this latter proof here. We have that the rows ai1 , . . . , air , ak are linearly dependent, while the a , . . . , air are linearly independent. If the rows ai2 , . . . , air , ak are linearly independent, then, from Proposition 1.16, the rank of the matrix composed from the rows indexed i1 , . . . , ir , ir+1 , k is exactly r. But, by assumption, the ai1 , . . . , air+1 are linearly independent. Thus the ai2 , . . . , air , ak are necessarily linearly dependent. We repeat this argument, each time lopping off the first vector, until we arrive at the desired result that ak is linearly dependent, i.e., ak is the zero vector. i1

The above results allow us to totally characterize the possible vanishing minors of a nonsingular totally positive matrix. We have the following. Theorem 1.19 Let A be an n × n nonsingular totally positive matrix. Let (αk , βk , rk )k=1 be the set of all triples such that for each k ∈ {1, . . . , } A

αk + 1, . . . , αk + rk βk + 1, . . . , βk + rk

= 0,

and no principal minors of the rk × rk submatrix αk + 1, . . . , αk + rk A βk + 1, . . . , βk + rk


19

vanish. For such submatrices we have αk = βk . Moreover i1 , . . . , ip =0 A j1 , . . . , jp if and only if for some k ∈ {1, . . . , } there is an rk ×rk principal submatrix of i1 , . . . , ip A j1 , . . . , jp that lies in the right shadow of an above αk + 1, . . . , αk + rk A βk + 1, . . . , βk + rk if αk < βk , or in its left shadow if αk > βk . In other words, what we are proving here is that for a nonsingular totally positive matrix there are certain basic zero minors formed from consecutive rows and columns and all other zero minors are derived from them. This derivation is also of a very specific nature. It is a consequence of a vanishing principal minor of the zero minor lying in the shadow of one of these basic zero minors. Proof Assume

A

and no principal minor of

αk + 1, . . . , αk + rk βk + 1, . . . , βk + rk

= 0,

αk + 1, . . . , αk + rk A βk + 1, . . . , βk + rk

vanishes. As A is nonsingular it follows from Theorem 1.13 that αk = βk . In addition, from Proposition 1.15 we have that each such vanishing minor either throws a right or a left shadow. If αk < βk then it must throw a right shadow since the left shadow of αk + 1, . . . , αk + rk A βk + 1, . . . , βk + rk is

αk + 1, . . . , n A 1, . . . , βk + rk

which contains the nonsingular rk × rk principal submatrix αk + 1, . . . , αk + rk . A αk + 1, . . . , αk + rk

20


But this contradicts Theorem 1.13. Similarly if αk > βk then αk + 1, . . . , αk + rk A βk + 1, . . . , βk + rk must throw a left shadow. Now if

i1 , . . . , ip A j1 , . . . , jp

is such that it contains an rk × rk principal submatrix that lies in the right shadow of one of the αk + 1, . . . , αk + rk A βk + 1, . . . , βk + rk if αk < βk , or in its left shadow if αk > βk , then from Proposition 1.15 this principal minor of i1 , . . . , ip A j1 , . . . , jp must vanish. It now follows from Theorem 1.13 that we must have i1 , . . . , ip A = 0. j1 , . . . , jp This proves the easier direction of the theorem. Let us now assume that i1 , . . . , ip A =0 j1 , . . . , jp for some choice of 1 ≤ i1 < · · · < ip ≤ n and 1 ≤ j1 < · · · < jp ≤ m. We assume, in what follows, that no principal minors of this minor vanish. (Otherwise replace the above by a principal minor with this same property.) As i1 , . . . , ip A =0 j1 , . . . , jp while

A

i1 , . . . , ip−1 j1 , . . . , jp−1

,A

i2 , . . . , ip j2 , . . . , jp

>0

it follows from Proposition 1.17 that the (ip − i1 + 1) × (jp − j1 + 1) totally positive matrix i1 , i1 + 1, . . . , ip A j1 , j1 + 1, . . . , jp (composed of consecutive rows and columns) has rank p − 1. In this proof


21

we can assume p > 1 as the case p = 1 is a direct consequence of Proposition 1.15. We claim that jp − i1 ≤ p − 2

or

ip − j1 ≤ p − 2.

Assume not. Then jp − i1 ≥ p − 1

and

ip − j1 ≥ p − 1.

(1.4)

Set α = max{i1 , j1 } − 1 We claim that i1 ≤ α + 1 < · · · < α + p ≤ ip and j1 ≤ α + 1 < · · · < α + p ≤ jp . To see this note that if α = i1 − 1, then α + p = i1 + p − 1 ≤ ip (since the i1 , . . . , ip are increasing integers), while α + 1 = i1 ≥ j1 and from (1.4) α + p = i1 + p − 1 ≤ jp . If α = j1 − 1 the same analysis applies. As i1 , i1 + 1, . . . , ip A j1 , j1 + 1, . . . , jp has rank p − 1, it therefore follows that α + 1, . . . , α + p A = 0, α + 1, . . . , α + p which contradicts Theorem 1.13. Thus jp − i1 ≤ p − 2

or

ip − j1 ≤ p − 2.

Let us assume that jp − i1 ≤ p − 2. The matrix i1 , i1 + 1, . . . , i1 − p + 1 A jp − p + 1, jp − p + 2, . . . , jp composed of p consecutive rows and columns is of rank, at most, p−1. Thus by Proposition 1.15 this submatrix throws a right or left shadow. From the analysis of the first part of the proof of this theorem we see that it throws a left shadow since i1 > jp − p + 1. That is, i1 , . . . , n A 1, . . . , jp

22


is of rank p − 1. As

i1 , . . . , ip A j1 , . . . , jp

lies in this submatrix we have that i1 , . . . , ip A j1 , . . . , jp lies in the left shadow of

A

α + 1, . . . , α + p β + 1, . . . , β + p

where α = i1 − 1 and β = jp − p. In fact from our assumption that no principal minors of i1 , . . . , ip A j1 , . . . , jp vanish, it necessarily follows that α + 1, . . . , α + p A β + 1, . . . , β + p is of rank exactly p − 1. The case where ip − j1 ≤ p − 2 is handled similarly. That is, it follows that the i1 , . . . , ip A j1 , . . . , jp is in the right shadow of the matrix ip − p + 1, ip − p + 2, . . . , ip A j1 , j1 + 1, . . . , j1 + p − 1 of rank p − 1. This proves the theorem. One simple application of Theorem 1.19 is the following. As we have seen from Theorem 1.13, if A is totally positive and i1 , . . . , ip >0 A j1 , . . . , jp then aik ,jk > 0, k = 1, . . . , p. In general the converse need not hold. We claim that if the converse holds for consecutive row and column indices, then it holds in general.


23

Proposition 1.20 Let A be an n × n nonsingular totally positive matrix. Assume that i + 1, . . . , i + p A >0 j + 1, . . . , j + p if ai+k,j+k > 0, k = 1, . . . , p, for all possible i, j and p. Then for all 1 ≤ i1 < · · · < ip ≤ n, 1 ≤ j1 < · · · < jp ≤ n, and all p we have i1 , . . . , ip >0 A j1 , . . . , jp if aik ,jk > 0, k = 1, . . . , p. Proof Assume

A

i1 , . . . , ip j1 , . . . , jp

= 0.

From Theorem 1.19 there exist (α, β, r) such that α + 1, . . . , α + r A = 0, β + 1, . . . , β + r no principal submatrix of

A

α + 1, . . . , α + r β + 1, . . . , β + r

vanishes, and some r × r principal submatrix of i1 , . . . , ip A j1 , . . . , jp lies in the right shadow of

α + 1, . . . , α + r A β + 1, . . . , β + r

if α < β, or in its left shadow if α > β. The assumption of the proposition implies that r = 1 and thus aα+1,β+1 = 0. We therefore have an α = β such that for some k ∈ {1, . . . , p} the aik ,jk lies in the right shadow of aα+1,β+1 = 0 if α < β, or in its left shadow if α > β. This implies that aik ,jk = 0. Totally positive matrices that satisfy the above property, i.e., i1 , . . . , ip >0 A j1 , . . . , jp

24


if and only if aik ,jk > 0,

k = 1, . . . , p,

are called almost strictly totally positive matrices.

1.4 Determinantal inequalities There are various determinantal inequalities that hold for totally positive and strictly totally positive matrices. In this section we detail a few of them. Generalized Hadamard Inequalities We start with the generalized Hadamard inequalities that hold for a wide class of matrices. In this monograph we prove these inequalities only for totally positive matrices. Let i, j, and k be disjoint sets in {1, . . . , n}. We prove the following, where we slightly abuse notation. Theorem 1.21 Let A be any n × n totally positive matrix and i, j, and k be disjoint subsets of {1, . . . , n}. Then, A(i, j, k)A(i) ≤ A(i, j)A(i, k).

(1.5)

If A is strictly totally positive, then strict inequality holds in (1.5). We recall what we mean by the above inequality. If i = (i1 , . . . , ip ) ∈ Ipn , j = (j1 , . . . , jq ) ∈ Iqn , k = (k1 , . . . , kr ) ∈ Irn are disjoint sets of ordered indices in {1, . . . , n}, then the following inequalities hold between the principal minors of A: i1 , . . . , ip i1 , . . . , ip , j1 , . . . , jq , k1 , . . . , kr A A i1 , . . . , ip , j1 , . . . , jq , k1 , . . . , kr i1 , . . . , ip i1 , . . . , ip , k1 , . . . , kr i1 , . . . , ip , j1 , . . . , jq ≤A A , i1 , . . . , ip , j1 , . . . , jq i1 , . . . , ip , k1 , . . . , kr where we (always) assume that the row and column indices have been rearranged in natural (increasing) order. This inequality is called a Generalized Hadamard Inequality. It is sometimes written A(s ∪ t)A(s ∩ t) ≤ A(s)A(t) where s and t are arbitrary subsets of {1, . . . , n}. In our notation s = i ∪ j and t = i ∪ k. If s ∩ t = i = ∅, then (1.5) remains valid and is generally called a Hadamard Inequality. (For convenience we always set A(∅) = 1.)

1.4 Determinantal inequalities

25

Proof We prove the result for a totally positive matrix. If A(i, j, k) = 0 then (1.5) is certainly valid. As such we may assume that A(i, j, k) > 0. Thus from Theorem 1.13 all the other minors in (1.5) are strictly positive. Let i, k . bk = A i, Set |j| = q and |k| = r, where | · | denotes the cardinality of the set. From Sylvester’s Determinant Identity B(j) = A(i, j)A(i)q−1 B(k) = A(i, k)A(i)r−1 and B(j, k) = A(i, j, k)A(i)q+r−1 . Thus (1.5) is equivalent to the simpler Hadamard inequality B(j, k) ≤ B(j)B(k),

(1.6)

where j and k are disjoint sets of ordered indices in {1, . . . , n}, B is a totally positive matrix, and B(j, k) > 0. It is this inequality that we now prove. Our proof will be by induction on q + r = m. We always assume that q, r ≥ 1. For m = 2 we must show that k j j, k = bjj bkk B ≤B B k j j, k where j = (j) and k = (k). As j, k B = bjj bkk − bjk bkj j, k and bjk , bkj ≥ 0, this inequality is immediate. Assume m > 2 and q > 1. Let j1 ∈ j and j = j\{j1 } and set j1 , k ck = B j1 , for k, ∈ j ∪ k. From Sylvester’s Determinant Identity, C = (ck ) is totally positive and C(j , k) > 0. Now C(j ) = B(j)bq−2 j1 j1 C(k) = B(k, j1 )bjr−1 1 j1

26


and . C(j , k) = B(j, k)bjq+r−2 1 j1 From the induction hypothesis and since |j | + |k| = m − 1 C(j , k) ≤ C(j )C(k), implying, by the above set of equalities, B(j, k)bj1 j1 ≤ B(j)B(k, j1 ). Since |k| + 1 = r + 1 < m, we can again apply the induction hypothesis to obtain B(k, j1 ) ≤ B(k)bj1 j1 . Substituting in the above gives us the desired inequality (1.6). This same proof, with minor modifications in order to verify strict inequality, is valid for strictly totally positive matrices. Remark In all the determinantal inequalities in this section we consider only principal minors. Recall (Proposition 1.2) that as all submatrices of (strictly) totally positive matrices are themselves (strictly) totally positive matrices these inequalities are also valid for nonprincipal minors. But be careful with the bookkeeping. There are various consequences of this generalized Hadamard inequalities. We present two of them here. Theorem 1.22 Let A be an n × n totally positive matrix. Let i1 , . . . , iq be arbitrary subsets of {1, . . . , n}. Set j = {r : r belongs to at least of the i1 , . . . , iq }. Then A(j1 ) · · · A(jq ) ≤ A(i1 ) · · · A(iq ). If A is strictly totally positive, then strict inequality holds in the above. Proof We prove this result for a totally positive matrix. The same proof, with minor modifications, is valid for strictly totally positive matrices. Our proof will be by induction on q. Note that the case q = 2 is exactly Theorem 1.21. Set k = {r : r belongs to at least of the i1 , . . . , iq−1 }.

1.4 Determinantal inequalities

27

From the induction hypothesis A(k1 ) · · · A(kq−1 ) ≤ A(i1 ) · · · A(iq−1 ).

(1.7)

Let m = {r : r belongs to iq and at least of the i1 , . . . , iq−1 }. Thus j = k ∪ m−1 and m = k ∩ m−1 ,

= 1, . . . , q,

where m0 = iq , mq = kq = ∅ and A(∅) = 1. Applying Theorem 1.21 we obtain A(j )A(m ) ≤ A(k )A(m−1 ),

= 1, . . . , q.

Thus q

A(j )A(m ) ≤

=1

q

A(k )A(m−1 ),

=1

which reduces to q

q−1

A(j ) ≤

=1

A(k ) A(m0 )

=1

and from (1.7) this implies that q−1

q q A(j ) ≤ A(i ) A(iq ) = A(i ). =1

=1

=1

Let A be an n × n matrix and for r = 1, . . . , n, set  1  (n−1 i1 , . . . , ir  r−1 ) Qr =  A . i1 , . . . , ir 1≤i1 0, and assume the result holds for all i with d(i) < r. As the sequence i is not composed of consecutive integers, we can add to this sequence an integer between i1 and im . Let 1 ≤ j1 < · · · < jm+1 ≤ n be the resulting sequence assuming j to be the new index. Consider the determinantal identity (1.2) from Chapter 1:

j1 , . . . , j , . . . , jm+1 j2 , . . . , jm C C 1, . . . , m − 1 1, . . . , m

j2 , . . . , jm+1 j1 , . . . , j , . . . , jm = C C 1, . . . , m − 1 1, . . . , m

j2 , . . . , j1 , . . . , jm j , . . . , jm+1 +C C . 1, . . . , m − 1 1, . . . , m In the above equation the three m − 1 order minors are all positive by assumption, as they are all based on the columns 1, . . . , m − 1. Now d(i) = (im − i1 ) − (m − 1) = (jm+1 − j1 ) − (m − 1) = r, while (jm+1 − j2 ) − (m − 1) and (jm − j1 ) − (m − 1) are strictly less than r. Thus, by our induction hypothesis, we have j1 , . . . , jm j2 , . . . , jm+1 C , C > 0. 1, . . . , m 1, . . . , m This implies that

i1 , . . . , im j1 , . . . , j , . . . , jm+1 C =C > 0. 1, . . . , m 1, . . . , m We will use the method of proof of Fekete’s Lemma at least twice more in this chapter. An immediate consequence of the above result is the following. Theorem 2.2 Let A be an n×m matrix. Assume that all kth order minors of A composed from consecutive rows and consecutive columns are strictly positive for k = 1, . . . , min{n, m}. Then A is strictly totally positive.

2.1 Criteria for strict total positivity

39

However, we need not verify all these inequalities in order to determine if A is strictly totally positive. It suffices to check the strict positivity of an even smaller number of determinants. Theorem 2.3 Let A be an n×m matrix. Assume that all kth order minors of A composed from the first k rows and k consecutive columns, and also all kth order minors of A composed from the first k columns and k consecutive rows, are strictly positive for k = 1, . . . , min{n, m}. Then A is strictly totally positive. There are thus exactly nm determinantal inequalities to be checked. For n = m this n2 is a significantly smaller value than 4n n−1/2 . Proof We prove that the conditions in Theorem 2.3 imply that all kth order minors of A composed from consecutive rows and columns are strictly positive for k = 1, . . . , min{n, m}. We then apply Theorem 2.2 to obtain our desired result. In other words we prove that i + 1, . . . , i + k >0 (2.1) A j + 1, . . . , j + k for i = 0, . . . , n − k, j = 0, . . . , m − k, and k = 1, . . . , min{n, m}. Our assumption is that this result holds when i = 0 and when j = 0. Let us first prove that (2.1) also holds when j = 1, i.e., i + 1, . . . , i + k A > 0. 2, . . . , k + 1 We prove this by induction on i and on k. Assume i = 1. For k = 1 we must prove that 2 A = a22 > 0. 2 Now

A

1, 2 1, 2

= a11 a22 − a21 a12 > 0.

By assumption a11 , a21 , a12 > 0. Thus a22 > 0. Assume the result holds for k − 1. By an application of Sylvester’s Determinant Identity, 2, . . . , k + 1 1, . . . , k 2, . . . , k + 1 1, . . . , k A −A A A 1, . . . , k 2, . . . , k + 1 2, . . . , k + 1 1, . . . , k 2, . . . , k 1, . . . , k + 1 =A A . 2, . . . , k 1, . . . , k + 1

40

Criteria for total positivity and strict total positivity

All determinants containing initial consecutive rows or columns are strictly positive by assumption, and from the induction hypothesis 2, . . . , k > 0. A 2, . . . , k Thus

A

2, . . . , k + 1 2, . . . , k + 1

> 0.

Let us now consider the case i = 2. For k = 1 we must prove that a32 > 0. Now 2, 3 = a21 a32 − a31 a22 > 0. A 1, 2 By assumption a21 , a31 > 0. From the previous analysis a22 > 0. Thus a32 > 0. Assume the result holds for k − 1. From Sylvester’s Determinant Identity, 2, . . . , k + 1 3, . . . , k + 2 2, . . . , k + 1 3, . . . , k + 2 A A −A A 1, . . . , k 2, . . . , k + 1 2, . . . , k + 1 1, . . . , k 3, . . . , k + 1 2, . . . , k + 2 = A A . 2, . . . , k 1, . . . , k + 1 All determinants containing initial consecutive rows or columns are strictly positive, and by the above case i = 1 2, . . . , k + 1 > 0. A 2, . . . , k + 1 From the induction hypothesis 3, . . . , k + 1 > 0. A 2, . . . , k Thus

A

3, . . . , k + 2 2, . . . , k + 1

> 0.

We continue this process for all possible i. We do the same for i = 1 and all j and k. We then go through the argument again for j = 2, then i = 2, etc. . . , or alternatively, since the above result now holds for i = 1 and j = 1, we can apply an induction argument to the (n − 1) × (m − 1) matrix obtained from A by deleting its first row and column. In Section 2.4 of this chapter we give a different and more transparent proof of this result.

2.2 Density and some further applications

41

When dealing with strictly totally positive matrices we can reverse the order of the rows and columns (see Proposition 1.3). As a consequence thereof, we have the following. Corollary 2.4 Let A be an n×m matrix. Assume that all kth order minors of A composed from the last k rows and k consecutive columns, and also all kth order minors of A composed from the last k columns and k consecutive rows, are strictly positive for k = 1, . . . , min{n, m}. Then A is strictly totally positive. Similar corollaries hold for many of the results of this chapter. They should be understood. We will not bother to state them again. Is this the minimal number of determinants that must be checked when determining whether A is a strictly totally positive matrix? It seems that it must be minimal as the number of conditions is nm, which is the number of entries in the matrix A. The following is presented here as it fits in with our present discussion and can be seen to be a simple consequence of Theorem 2.3. It is also an immediate corollary of Theorem 1.19. We state it without proof. Proposition 2.5 Let A be an n × n totally positive matrix. Then A is strictly totally positive if 1, . . . , k > 0 A n − k + 1, . . . , n n − k + 1, . . . , n > 0 A 1, . . . , k for k = 1, . . . , n.

2.2 Density and some further applications Strictly totally positive matrices are easier to handle than totally positive matrices and fortuitously the class of strictly totally positive matrices is dense in the class of all totally positive matrices. That is, if A is an n × m totally positive matrix, then there are sequences (Ak )∞ k=1 of n × m strictly totally positive matrices such that elementwise lim Ak = A.

k→∞

This result will prove very useful.

42


Theorem 2.6 Strictly totally positive matrices are dense in the class of totally positive matrices. 2

Proof For each q ∈ (0, 1) the matrix Qn (q) = (q (i−j) )ni=1 nj=1 is a strictly 2 totally positive matrix. This may be proven as follows. As q (i−j) = 2 2 q i q j q −2ij it suffices to prove that pij is strictly totally positive for p = q −2 > 1. Let P = (pij )ni=1 m j=1 . Consider i + 1, . . . , i + k . P j + 1, . . . , j + k Set xr = pi+r , r = 1, . . . , k. It then easily follows, factoring out from the rth row of this minor p(i+r)(j+1) , that

k i + 1, . . . , i + k (i+r)(j+1) p V (x1 , . . . , xk ) , = P j + 1, . . . , j + k r=1 where V (x1 , . . . , xk ) is the Vandermonde of the points x1 , . . . , xk and thus equals (xt − xs ) = (pi+t − pi+s ). 1≤s0 P j + 1, . . . , j + k and therefore, from Theorem 2.2, the matrix Qn (q) is strictly totally positive. The matrix Qn (q) has the further property that lim Qn (q) = In

q→0+

(elementwise) where In is the n × n identity matrix. Now, let A be an arbitrary n × m totally positive matrix of rank r. Form Bq = Qn (q)AQm (q). Then each Bq is of rank r, and as is easily verified using the Cauchy–Binet formula, all the k × k minors of Bq are strictly positive for every k ≤ r. Furthermore lim Bq = A.

q→0+

If r = min{n, m} we are finished. Assume not and let Bq be identical with the matrix Bq except that we add an arbitrarily small ε > 0 to the


43

(1, 1)-element of Bq . The matrix Bq is totally positive and of rank r + 1. As such, by the same method as employed above, we can approximate Bq arbitrarily well by a matrix of rank r + 1, all of whose k × k minors are strictly positive for every k ≤ r + 1. We continue this process to obtain the result. Determinantal criteria for when a matrix is totally positive will be discussed in Section 2.4. But we present here a particular case of Theorem 2.13 since it can be easily proven by an application of Theorem 2.2 and the method of proof of Theorem 2.6. This is not the best possible result, as will be shown in Proposition 2.15. Proposition 2.7 Let A be an n × n nonsingular matrix. Assume that all kth order minors of A composed from arbitrary k rows and k consecutive columns, or all kth order minors of A composed from arbitrary k columns and k consecutive rows are nonnegative for k = 1, . . . , n. Then A is totally positive. Proof Assume all kth order minors of A composed from arbitrary k rows and k consecutive columns are nonnegative. Let Q(q) be the n × n matrix as used in the proof of Theorem 2.6, q ∈ (0, 1). Set Aq = Q(q)A. From the Cauchy–Binet formula i + 1, . . . , i + k Aq j + 1, . . . , j + k 1 , . . . , k i + 1, . . . , i + k = A . Q(q) 1 , . . . , k j + 1, . . . , j + k 1≤1 0, Aq j + 1, . . . , j + k

44


for all i, j ∈ {0, . . . , n−k}. From Theorem 2.2 Aq is strictly totally positive. In addition, lim Aq = A,

q→0+

and therefore A is totally positive. Theorems 2.2 and 2.3 are very useful tools for proving strict total positivity. We now settle a previous debt by presenting proofs of Propositions 1.9 and 1.10 from Section 1.2. We recall the following. Proposition 1.9 Assume A is an n × m strictly totally positive matrix. For given 1 ≤ r < n set aij, i = 1, . . . , r, j = 2, . . . , m cij = i−1,i A 1,j , i = r + 1, . . . , n, j = 2, . . . , m. Then C = (cij ) is an n × (m − 1) strictly totally positive matrix. If A is totally positive, then C is totally positive. Proof Let C = (cij ) be as above. We prove our desired result by obtaining an explicit formula for the arbitrary minors of C composed of consecutive rows, i.e., i + 1, . . . , i + k . C j1 , . . . , jk The first statement in Proposition 1.9 will then follow from either Theorem 2.2 or 2.3. Obviously if i + k ≤ r, then i + 1, . . . , i + k i + 1, . . . , i + k =A . C j1 , . . . , jk j1 , . . . , jk We prove that for i + 1 ≤ r < i + k i + 1, . . . , i + k i + 1, . . . , i + k = ar,1 · · · ai+k−1,1 A C j1 , . . . , jk j1 , . . . , jk and for r < i + 1 i, i + 1, . . . , i + k i + 1, . . . , i + k = ai+1,1 ai+2,1 · · · ai+k−1,1 A . C 1, j1 , . . . , jk j1 , . . . , jk


45

The first equality can be explained as follows.

C

i + 1, . . . , i + k j1 , . . . , jk

=

ai+1,j1 .. .

···

ai+1,jk .. .

ar,j1

···

ar,jk

A A

···

r,r+1 1,j1

.. . i+k−1,i+k 1,j1

A

···

A

.

r,r+1 1,jk

.. . i+k−1,i+k 1,jk

By adding ar+1,1 times the (r − i)th row (which is the row with entries (ar,j1 , . . . , ar,jk )) to the succeeding row, we get a new (r − i + 1)st row of (ar,1 ar+1,j1 , . . . , ar,1 ar+1,jk ). Factor out ar,1 and add ar+2,1 times this row to the next row to obtain a new (r − i + 2)nd row of (ar+1,1 ar+2,j1 , . . . , ar+1,1 ar+2,jk ), etc. The formula now easily follows. The second equality is similarly obtained by considering the determinant ai1 0 .. . 0

···

aij1 A A

.. . i+k−1,i+k 1,j1

aijk

···

i,i+1 1,j1

···

A A

i,i+1 1,jk

.. . i+k−1,i+k 1,jk

.

On the one hand this determinant equals i + 1, . . . , i + k . ai1 C j1 , . . . , jk On the other hand, adding ai+1,1 times the first row to the second row we get a new second row of (ai1 ai+1,1 , ai1 ai+1,j1 , . . . , ai1 ai+1,jk ). Factor out ai1 and add ai+2,1 times the second row to the third row etc. The formula now easily follows. From these three formulæ it follows from Theorems 2.2 or 2.3 that if A is strictly totally positive then C is strictly totally positive. Now assume that A is only totally positive. From Theorem 2.6 there exists a sequence (Ak ) of n × m strictly totally positive matrices whose elements converge

46


to the corresponding elements of A. Construct the associated Ck (in the same way in which C is constructed from A). Each Ck is strictly totally positive and the elements of Ck converge to the corresponding elements of C. Thus C is totally positive. Proposition 1.10 Assume A is an n × m strictly totally positive matrix. Fix p < min{m, n} and set i − p, . . . , i − 1, i cij = A 1, . . . , p, j for i = p + 1, . . . , n, j = p + 1, . . . , m. Then C = (cij ) is an (n − p) × (m − p) strictly totally positive matrix. Proof We prove Proposition 1.10 as a consequence of Proposition 1.9. As A is a strictly totally positive matrix, Proposition 1.10 holds for p = 1 since the associated C is a submatrix of the matrix C in Proposition 1.9 with r = 1. We apply an induction argument on p. Let us assume the validity of Proposition 1.10 for p − 1. That is, setting i − p + 1, . . . , i − 1, i eij = A 1, . . . , p − 1, j for i = p, . . . , n, j = p, . . . , m, we assume that E = (eij ) is an (n − p + 1) × (m − p + 1) strictly totally positive matrix. Now define i − 1, i dij = E p, j for i = p + 1, . . . , n, j = p + 1, . . . , m. Applying Proposition 1.9 with r = 1 it follows that D = (dij ) is an (n − p) × (m − p) strictly totally positive matrix. In addition, from Sylvester’s Determinant Identity, i−p,...,i−2,i−1 i−p,...,i−2,i−1 A A i − 1, i 1,...,p−1,j 1,...,p−1,p = dij = E i−p+1,...,i−1,i p, j A i−p+1,...,i−1,i A 1,...,p−1,p 1,...,p−1,j i − p + 1, . . . , i − 1 i − p, i − p + 1, . . . , i − 1, i = A A . 1, . . . , p − 1 1, . . . , p − 1, p, j Note that the values A

i − p + 1, . . . , i − 1 1, . . . , p − 1

2.3 Triangular total positivity

47

are strictly positive and independent of j. Thus the matrix D being strictly totally positive is equivalent to the matrix C with elements i − p, i − p + 1, . . . , i − 1, i cij = A 1, . . . , p − 1, p, j being strictly totally positive. This advances the induction step and proves the proposition.

2.3 Triangular total positivity A square matrix A = (aij ) is said to be upper triangular if aij = 0 for all i > j. It is said to be lower triangular if aij = 0 for all i < j. We say that the n × n matrix A = (aij ) is upper strictly totally positive if it is upper triangular and i1 , . . . , ik >0 A j1 , . . . , jk whenever im ≤ jm , m = 1, . . . , k. (As usual we always assume that 1 ≤ i1 < · · · < ik ≤ n and 1 ≤ j1 < · · · < jk ≤ n.) Note that for an upper triangular matrix we always have i1 , . . . , ik A =0 j1 , . . . , jk if im > jm for some m ∈ {1, . . . , k}. We say that the n×n matrix A = (aij ) is upper totally positive if it is upper triangular and i1 , . . . , ik A ≥0 j1 , . . . , jk whenever im ≤ jm , m = 1, . . . , k. (This latter restriction has no meaning here.) Similarly we say that the n × n matrix A = (aij ) is lower strictly totally positive if it is lower triangular and i1 , . . . , ik A >0 j1 , . . . , jk whenever im ≥ jm , m = 1, . . . , k. We say that the n × n matrix A = (aij ) is lower totally positive if it is lower triangular and i1 , . . . , ik A ≥0 j1 , . . . , jk whenever im ≥ jm , m = 1, . . . , k. There are relatively simple criteria for determining when a triangular

48


matrix is strictly totally positive in the above sense. These criteria are very reminiscent of the criteria in Theorem 2.3, and for good reason. Theorem 2.8 Let A = (aij ) be an n × n upper triangular matrix satisfying 1, . . . , k >0 (2.2) A j + 1, . . . , j + k for j = 0, 1, . . . , n−k, k = 1, . . . , n. Then A is upper strictly totally positive. Similarly, if A = (aij ) is an n × n lower triangular matrix satisfying i + 1, . . . , i + k >0 A 1, . . . , k for i = 0, 1, . . . , n − k, k = 1, . . . , n, then A is lower strictly totally positive. Proof Obviously the two claims are equivalent. We prove the first claim. That is, we show that if A is an n × n upper triangular matrix satisfying (2.2), then i1 , . . . , ik A >0 j1 , . . . , jk whenever im ≤ jm , m = 1, . . . , k. From (2.2) and Fekete’s Lemma 2.1, it follows that 1, . . . , k >0 A j1 , . . . , jk

(2.3)

for all 1 ≤ j1 < · · · < jk ≤ n, and k = 1, . . . , n. Since A is upper triangular and satisfies (2.3) we also have that for any 1 ≤ j1 < · · · < jk ≤ n, where i + m ≤ jm , m = 1, . . . , k, i + 1, . . . , i + k 1, . . . , i 1, . . . , i, i + 1, . . . , i + k A A =A > 0. j1 , . . . , jk 1, . . . , i 1, . . . , i, j1 , . . . , jk This implies that

A

i + 1, . . . , i + k j1 , . . . , jk

>0

(2.4)

for any 1 ≤ j1 < · · · < jk ≤ n, where i + m ≤ jm , m = 1, . . . , k. It thus remains to prove that we can replace consecutive rows by arbitrary rows (within the confines of the upper triangularity). To this end we rework the method of proof of Fekete’s Lemma 2.1. For

2.3 Triangular total positivity

49

a given i = (i1 , . . . , ik ) ∈ Ikn we recall that d(i), the dispersion of i, is given by d(i) :=

k

(ij − ij−1 − 1) = (ik − i1 ) − (k − 1) ≥ 0.

j=2

Then d(i) = 0 if and only if the sequence i is composed of consecutive integers. We prove the result by induction on both k and on the value of d(·). We assume that i1 , . . . , is >0 A j1 , . . . , js whenever im ≤ jm , m = 1, . . . , s, and s ≤ k and d(i) < p. From (2.4) the result is valid for every k when d(i) = 0. Let i = (i1 , . . . , ik ) with d(i) = p > 0, and im ≤ jm , m = 1, . . . , k. We claim that i1 , . . . , ik > 0. A j1 , . . . , jk As the sequence i is not composed of consecutive integers, we can add to this sequence an integer between i1 and ik . Let 1 ≤ r1 < · · · < rk+1 ≤ n be the resulting sequence, assuming r to be the new index (1 < < k + 1). Thus rm

≤

jm ,

m = 1, . . . , − 1

rm < rm+1

≤

jm ,

m = , . . . , k.

(2.5)

By the determinantal identity (1.2) of Section 1.1, we have r2 , . . . , rk r1 , . . . , r , . . . , rk+1 A A j1 , . . . , jk j2 , . . . , jk r1 , . . . , r , . . . , rk r2 , . . . , rk+1 = A A j1 , . . . , jk j2 , . . . , jk r2 , . . . , r , . . . , rk+1 r1 , . . . , rk +A A . j1 , . . . , jk j2 , . . . , jk Now

A

r2 , . . . , rk j2 , . . . , jk

>0

by the induction assumption on k and since rm ≤ jm , m = 2, . . . , k (see (2.5)), while r1 , . . . , rk A >0 j1 , . . . , jk

50


since rm ≤ jm , m = 1, . . . , k (see (2.5)) and by the induction assumption on d(i). That is, since r1 = i1 and rk < ik it follows that d((r1 , . . . , rk )) < d(i). Furthermore r2 , . . . , r , . . . , rk+1 A >0 j2 , . . . , jk by the induction hypothesis on k and since rm ≤ jm , m = 2, . . . , − 1, and rm+1 ≤ jm , m = , . . . , k (see (2.5)). Similarly we have that r1 , . . . , r , . . . , rk A > 0. j2 , . . . , jk Now

A

r2 , . . . , rk+1 j1 , . . . , jk

≥ 0,

because this determinant vanishes if we have rm+1 > jm for some m ∈ {1, . . . , −1} (which is possible), while if not then it is strictly positive since d((r2 , . . . , rk+1 )) < d(i). It therefore follows that r1 , . . . , r , . . . , rk+1 i1 , . . . , ik A =A >0 j1 , . . . , jk j1 , . . . , jk which advances the induction hypothesis and proves the theorem. In parallel with Proposition 2.5 we have the following result, which we state for upper triangular matrices. It is also an immediate consequence of Theorem 1.19. Proposition 2.9 Let A be an n × n upper totally positive matrix. Then A is upper strictly totally positive if 1, . . . , k A >0 n − k + 1, . . . , n for k = 1, . . . , n.

2.4 LDU factorizations Assume A = (aij ) is an n × n strictly totally positive matrix. Since 1, . . . , k A >0 1, . . . , k

2.4 LDU factorizations

51

for k = 1, . . . , n, then, as is well known, A has a unique LU factorization. In fact A can be decomposed into the form A = LDU, where L = (ij ) is a unit diagonal lower triangular matrix, D = (dij ) is a diagonal matrix, and U = (uij ) is a unit diagonal upper triangular matrix. The nonzero elements of L, D and U are explicitly given by 1, . . . , j − 1, i A 1, . . . , j − 1, j , i ≥ j, ij = 1, . . . , j − 1, j A 1, . . . , j − 1, j 1, . . . , i − 1, i A 1, . . . , i − 1, j , uij = i ≤ j, 1, . . . , i − 1, i A 1, . . . , i − 1, i 1, . . . , i − 1, i A 1, . . . , i − 1, i . dii = 1, . . . , i − 1 A 1, . . . , i − 1 Using Theorem 2.8 we prove that L is a lower strictly totally positive matrix and U is an upper strictly totally positive matrix. Theorem 2.10 Assume A is a square strictly totally positive matrix. Then A has a unique factorization of the form A = LDU, where L is a unit diagonal lower strictly totally positive matrix, U is a unit diagonal upper strictly totally positive matrix, and D is a diagonal matrix whose diagonal entries are strictly positive. Proof We need only prove that L is a lower strictly totally positive matrix and U is an upper strictly totally positive matrix. The other facts follow trivially. As A = LDU and D is a diagonal matrix, then from the Cauchy–Binet formula 1, . . . , k A j + 1, . . . , j + k 1 , . . . , k 1 , . . . , k 1, . . . , k = . L D U 1 , . . . , k 1 , . . . , k j + 1, . . . , j + k 1≤1 0.

> 0.

As this is valid for all appropriate j and k we have from Theorem 2.8 that U is an upper strictly totally positive matrix. The analogous proof shows that L is a lower strictly totally positive matrix. Each square strictly totally positive matrix also has a unique factorization of the form A = U DL, where U is a unit diagonal upper strictly totally positive matrix, L is a unit diagonal lower strictly totally positive matrix, and D is a diagonal matrix whose diagonal entries are strictly positive. The nonzero elements of U , D, and L are explicitly given by i, j + 1, . . . , n A j, j + 1, . . . , n , uij = i ≤ j, j, j + 1, . . . , n A j, j + 1, . . . , n i, i + 1, . . . , n A j, i + 1, . . . , n , i ≥ j, ij = i, i + 1, . . . , n A i, i + 1, . . . , n i, i + 1, . . . , n A i, i + 1, . . . , n . dii = i + 1, . . . , n A i + 1, . . . , n

2.4 LDU factorizations This can be shown in a variety of ways. previous result and the following. Let  0 ··· 0  0 ··· 1  Q= . ..  .. . 1 ···

53

It also follows easily from the 1 0 .. .

   . 

0 0

Then QAQ is the matrix obtained from A by reversing the order of its rows and columns. As is known (see Proposition 1.3 in Chapter 1), A is strictly totally positive if and only if QAQ is strictly totally positive. From the LDU factorization of QAQ (Theorem 2.10) we have A = (QLQ)(QDQ)(QU Q). QLQ is a unit diagonal upper strictly totally positive matrix while QU Q is a unit diagonal lower strictly totally positive matrix. Another consequence of Theorem 2.10 is an alternative, more elementary, proof of Theorem 2.3. Another Proof of Theorem 2.3. Since 1, . . . , k A >0 1, . . . , k for k = 1, . . . , n, it follows that A has a unique LDU factorization. Since i + 1, . . . , i + k 1, . . . , k >0 ,A A 1, . . . , k j + 1, . . . , j + k for all appropriate j, i, and k, it follows from the proof of Theorem 2.10 that L is a unit diagonal lower strictly totally positive matrix, U is a unit diagonal upper strictly totally positive matrix, while D is a diagonal matrix whose diagonal entries are strictly positive. From this it easily follows that A = LDU is strictly totally positive. What about totally positive matrices? Do they always have a similar unique LDU factorization? If A is also non-singular, then the answer is yes. The proof is essentially the same as the above proof, except that L is a unit diagonal lower totally positive matrix and U is a unit diagonal upper totally positive matrix. Recall that since A is totally positive and nonsingular then

54


all principal minors of A are strictly positive (Theorem 1.13). We state this result here for easy reference. Proposition 2.11 Assume A is a square nonsingular totally positive matrix. Then A has a unique factorization of the form A = LDU where L is a unit diagonal lower totally positive matrix, U is a unit diagonal upper totally positive matrix, and D is a diagonal matrix whose diagonal entries are strictly positive. However, if A is singular then we must modify our demands somewhat. In general we lose the unit triangular nature of the L and U and the uniqueness. What we do have is the following. Theorem 2.12 Assume A is a square totally positive matrix. Then A has a factorization of the form A = LU where L is a lower totally positive matrix and U is an upper totally positive matrix. Proof From Theorem 2.6 there exists a sequence of strictly totally positive matrices (Am ) that approximate A, i.e., such that lim Am = A.

m→∞

From Theorem 2.10, each Am = (aij (m)) has a unique factorization of the form Am = Lm Dm Um where Lm is a unit diagonal lower strictly totally positive matrix, Um is a unit diagonal upper strictly totally positive matrix, and Dm is a diagonal matrix whose diagonal entries are strictly positive. There are many different normalizations to this factorization that allow us to take limits. Here is one such normalization. By pre- and postmultiplying by positive diagonal matrices we can factor Am in the form ! mL !m U !m Am = D ! m and U !m are stochastic (rows sums 1). As a consequence we also where L

2.5 Criteria for total positivity have

55

aij (m) = d!ii (m)

j

! m . Thus all entries in each for all i, where d!ii (m) is the (i, i) entry of D of the factors are uniformly bounded. Now that we have bounds on the ! m, L ! m and U !m , we may take a subsequence along which each elements of D of the entries of these matrices converges. Let their respective limits be D, and U , and set L = D L. Then L A = LU. Furthermore it follows that L is a lower totally positive matrix and U is an upper totally positive matrix.

2.5 Criteria for total positivity Unfortunately, nothing quite as simple as the conditions in Theorems 2.2 and 2.3 seem to be valid for determining if a matrix is totally positive. It is possible, for example, that every minor composed from consecutive rows and consecutive columns is nonnegative and yet A is not totally positive. A simple example is the 4 × 4 matrix   1 0 0 0  0 1 0 0   A  0 0 1 0 . 1 0 0 1 One therefore needs somewhat more restrictive criteria. Let us again recall that for a given strictly increasing sequence i = (i1 , . . . , ik ) ∈ Ikn we define d(i), the dispersion of i, by d(i) :=

k

(ij − ij−1 − 1) = (ik − i1 ) − (k − 1) ≥ 0.

j=2

Note that d(i) counts the number of integers between i1 and ik that are not in the sequence i1 , . . . , ik , and thus d(i) = 0 if and only if the sequence i is composed of consecutive integers. The main theorem providing determinantal criteria for when a matrix is totally positive is the following. It is based on previous reasonings and techniques, but is technically more detailed.

56


Theorem 2.13 An n × m matrix A = (aij ) of rank r is totally positive if A

i1 , . . . , ik j1 , . . . , jk

≥0

(2.6)

for all i = (i1 , . . . , ik ) satisfying d(i) ≤ n − r, all j = (j1 , . . . , jk ) and k = 1, . . . , r. Note that if A is nonsingular, i.e., r = n = m, then this is Proposition 2.7. Proof We wish to prove that from (2.6) it follows that A

i1 , . . . , is j1 , . . . , js

≥0

(2.7)

for all 1 ≤ i1 < · · · < is ≤ n and 1 ≤ j1 < · · · < js ≤ m and all s. The proof will be based on an induction argument. The induction is on s. For s = 1 there is nothing to prove since d({i1 }) = 0 ≤ n − r and thus (2.7) follows from (2.6). Let us suppose that (2.7) holds for all s ≤ k − 1, but is not valid for some (i1 , . . . , ik ) and (j1 , . . . , jk ), as above. That is, we have i1 , . . . , ik < 0. (2.8) A j1 , . . . , jk By our assumption, we must have d(i) > n − r. Choose i∗ = (i∗1 , . . . , i∗k ) and j∗ = (j1∗ , . . . , jk∗ ) such that (2.8) holds and d(i∗ ) = is minimal, i.e., for given k and every i ∈ Ikn , j ∈ Ikm satisfying (2.8) we have d(i) ≥ . If ai denotes the ith row of A, then we prove that for all p satisfying ∗ ∗ i∗1 < p < i∗k we have that ap ∈ span{ai2 , . . . , aik−1 }. As there are exactly such p (since d(i∗ ) = ), it follows that the rank of A is at most n − . But, by assumption, > n − r, which implies that the rank of A is strictly less than r. A contradiction. It therefore remains to prove that for all p satisfying i∗1 < p < i∗k we have ∗ ∗ ap ∈ span{ai2 , . . . , aik−1 }. This we prove in two steps. We first prove that it is true when we restrict ourselves to the columns j1∗ , . . . , jk∗ . We then subsequently extend it to all columns. Let p satisfy i∗1 < p < i∗k , and let r1 < · · · < rk+1 be the sequence obtained by adding p to the sequence i∗1 , . . . , i∗k . Assume p = rt . From the

2.5 Criteria for total positivity

57

determinantal equality (1.2) of Section 1.1, we have for each s ∈ {1, . . . , k}: r2 , . . . , rk r1 , . . . , r t , . . . , rk+1 A A j1∗ , . . . , jk∗ j1∗ , . . . , j ∗ , . . . , jk∗ s r1 , . . . , r t , . . . , rk r2 , . . . , rk+1 =A A (2.9) j1∗ , . . . , jk∗ j1∗ , . . . , js∗ , . . . , jk∗ r2 , . . . , r t , . . . , rk+1 r1 , . . . , rk +A ∗ A . j1 , . . . , jk∗ j1∗ , . . . , js∗ , . . . , jk∗ By the induction assumption, all three k − 1 × k − 1 minors in (2.9) are nonnegative, while the two k × k minors on the right-hand-side of the equation are nonnegative since the dispersion of the rows therein is strictly less than , and the minimality property of . Thus the value of the righthand-side of (2.9) is nonnegative. As (2.8) holds for i∗1 , . . . , i∗k and j1∗ , . . . , jk∗ (which is the left-most determinant in (2.9)), it necessarily follows that we must have

i∗2 , . . . , p, . . . , i∗k−1 r2 , . . . , rk A = 0. (2.10) =A ∗ j1 , . . . , js∗ , . . . , jk∗ j1∗ , . . . , js∗ , . . . , jk∗ ∗

∗

As (2.10) is valid for all s = 1, . . . , k, we have ap ∈ span{ai2 , . . . , aik−1 } on the columns j1∗ , . . . , jk∗ , for each p satisfying i∗1 < p < i∗k . Since ∗ i1 , . . . , i∗k A ∗ < 0, j1 , . . . , jk∗ ∗

∗

the rows ai1 , . . . , aik are linearly independent on the columns j1∗ , . . . , jk∗ . ∗ ∗ Thus the rows ai2 , . . . , aik−1 are linearly independent on some columns ! j1 , . . . , ! jk−2 , where {! j1 , . . . , ! jk−2 } = {j1∗ , . . . , jk∗ }\{js∗ , jt∗ } (we assume that ∗ ∗ js < jt in what follows). That is, ∗ i2 , . . . , i∗k−1 = 0. A ! jk−2 j1 , . . . , ! By our induction assumption, this implies that the above minor is, in fact, strictly positive. We use ∗ i2 , . . . , i∗k−1 A ! jk−2 j1 , . . . , ! as a pivot block in Sylvester’s Determinant Identity. That is, set ∗ i2 , . . . , i∗k−1 , i . dij = A ! jk−2 , j j1 , . . . , ! Then by our induction hypothesis on k we have dij ≥ 0 for all i and j,

58


and from (2.10) we have dpjs∗ = dpjt∗ = 0 for each p between i∗1 and i∗k . Furthermore, by Sylvester’s Determinant Identity we have ∗ ∗ ∗ ∗ i2 , . . . , i∗k−1 i1 , . . . , i∗k i1 , ik =A A ∗ < 0. D ! js∗ , jt∗ j1 , . . . , jk∗ j1 , . . . , ! jk−2 Thus we must have di∗1 jt∗ > 0 and di∗k js∗ > 0. From Sylvester’s Determinant Identity, for j < jt∗ , ∗ ∗ ∗ i2 , . . . , i∗k−1 i1 , . . . , i∗k−1 , p i1 , p =A A . D ! ! j, jt∗ jk−2 jk−2 , j, jt∗ j1 , . . . , ! j1 , . . . , ! Both terms of the right-hand side of this equality are nonnegative. The first of these two terms is strictly positive. The second term is nonnegative because the dispersion of i∗1 , . . . , i∗k−1 , p is strictly less than . Thus ∗ i ,p 0 ≤ D 1 ∗ = di∗1 j dpjt∗ − di∗1 jt∗ dpj . j, jt Now dpjt∗ = 0, di∗1 jt∗ > 0, and dpj ≥ 0. Thus dpj = 0 for all j < jt∗ . When j > js∗ we consider ∗ p, ik D js∗ , j and apply the same reasoning to obtain dpj = 0 for all j > js∗ . Thus for all j we have ∗ i2 , . . . , i∗k−1 , p . 0 = dpj = A ! jk−2 , j j1 , . . . , ! Since

i∗2 , . . . , i∗k−1 A ! j1 , . . . , ! jk−2 ∗

> 0, ∗

this implies that row ap is in span{ai2 , . . . , aik−1 } over all columns of A, and this is true for all p satisfying i∗1 < p < i∗k . One fairly elementary consequence of the above is the following result. Theorem 2.14 Let A = (aij ) be an n × n nonsingular upper triangular matrix satisfying 1, . . . , k A ≥0 j1 , . . . , jk for all 1 ≤ j1 < · · · < jk ≤ n, k = 1, . . . , n. Then A is upper totally

2.5 Criteria for total positivity

59

positive. Similarly, if A = (aij ) is an n × n nonsingular lower triangular matrix satisfying i1 , . . . , ik ≥0 A 1, . . . , k for 1 ≤ i1 < · · · < ik ≤ n, k = 1, . . . , n, then A is lower totally positive. Proof We prove the first claim. As A is a nonsingular upper triangular matrix we necessarily have amm > 0 for m = 1, . . . , n. From Theorem 2.13 it suffices to prove that i + 1, . . . , i + k ≥0 A j1 , . . . , jk for all appropriate i, 1 ≤ j1 < · · · < jk ≤ n and k (i.e., we must check the case d(i) = 0). It suffices to consider only those minors with i + m ≤ jm , m = 1, . . . , k, since all other minors vanish due to the upper triangular structure of A. Now, by assumption, 0≤A

1, . . . , i, i + 1, . . . , i + k 1, . . . , i, j1 , . . . , jk

=

i

amm A

m=1

i + 1, . . . , i + k j1 , . . . , jk

.

As amm > 0 for all m, this proves our result. Based on the above Theorem 2.14 and the analysis of Section 2.4 we have the following result, which should be compared to Theorem 2.13 and Proposition 2.7. Proposition 2.15 Let A = (aij ) be an n × n nonsingular matrix. Then A is totally positive if and only if A satisfies the following: 1, . . . , k (i) A 1, . . . , k > 0, k = 1, . . . , n i1 , . . . , ik 1 ≤ i1 < · · · < ik ≤ n, k = 1, . . . , n (ii) A 1, . . . , k ≥ 0, 1, . . . , k (iii) A j , . . . , j ≥ 0, 1 ≤ j1 < · · · < jk ≤ n, k = 1, . . . , n. 1

k

Proof If A is totally positive then (ii) and (iii) hold. If A is also nonsingular then (i) holds from Theorem 1.13. Assume (i) – (iii) hold. The LDU factorization of A, as in Section 2.4, is well defined since (i) holds. From (i), (ii), and (iii) we see that D is a diagonal matrix whose diagonal entries are strictly positive, L is a unit

60


diagonal lower triangular matrix, and U is a unit diagonal upper triangular matrix. Now, from the proof of Theorem 2.10 we have, 1 , . . . , k 1 , . . . , k i1 , . . . , ik i1 , . . . , ik = D U L A 1, . . . , k 1 , . . . , k 1 , . . . , k 1, . . . , k 1≤1 4 cos2 n+1 for all i, j = 1, . . . , n − 1. Then A is strictly totally positive. Furthermore π the above constant 4 cos2 n+1 is best possible.

2.6 “Simple” criteria for strict total positivity

61

By a simple perturbation argument and the denseness of strictly totally positive matrices in the class of all totally positive matrices it follows that if the inequality in Theorem 2.16 holds, but is not always strict, then A is totally positive. An immediate consequence of this theorem, independent of n and m, is the more easily stated. Corollary 2.17 Let A = (aij ) be an n×m matrix whose entries are strictly positive. Assume aij ai+1,j+1 ≥ 4ai,j+1 ai+1,j for all i = 1, . . . , n − 1, j = 1, . . . , m − 1. Then A is strictly totally positive. We start by proving that the constant 4 cos2 possible.

π n+1

in Theorem 2.16 is best

π there exists an n × n matrix Proposition 2.18 For any c < 4 cos2 n+1 A = (aij ) with all entries strictly positive for which

i, j = 1, . . . , n − 1,

aij ai+1,j+1 > c ai,j+1 ai+1,j , and yet A is not totally positive.

Proof Consider the n × n symmetric Toeplitz matrix  2 cos θ 1 0 ···  1 2 cos θ 1 ···   0 1 2 cos θ · ·· An (θ) =   . . . . .. .. .. ..  0

0

···

0

0 0 0 .. .

    ,  

2 cos θ

where 0 ≤ θ < π/2. Expanding by the last row we see that det An (θ) satisfies the recurrence relation det An (θ) = 2 cos θ det An−1 (θ) − det An−2 (θ). Now det A1 (θ) = 2 cos θ =

sin 2θ , sin θ

and det A2 (θ) = 4 cos2 θ − 1 =

sin 3θ . sin θ

62


From the fact that sin(n + 1)θ + sin(n − 1)θ = 2 cos θ sin nθ it immediately follows that det An (θ) =

sin(n + 1)θ , sin θ

implying that for θ ∈ (π/(n + 1), 2π/(n + 1)) we have det An (θ) < 0. For An (θ) = (aij ) we have aij ai+1,j+1 ≥ 4 cos2 θ ai,j+1 ai+1,j for i, j = 1, . . . , n − 1. Now for any c < 4 cos2

π n+1 ,

let

θ ∈ (π/(n + 1), 2π/(n + 1)) be such that c < 4 cos2 θ. Thus aij ai+1,j+1 ≥ c ai,j+1 ai+1,j for all i, j and yet det An (θ) < 0, i.e., An (θ) is far from being totally positive. The entries of An (θ) are not all strictly positive. But it is not a problem to perturb An (θ) so that its entries are all strictly positive, it satisfies aij ai+1,j+1 > c ai,j+1 ai+1,j for all i, j = 1, . . . , n − 1, and the perturbed An (θ) is not totally positive. In fact we can also do so while maintaining the Toeplitz character of the matrix. For example, keep the diagonal and first off-diagonal entries of 2 An (θ) fixed and perturb all elements of the kth off-diagonal to ε(k−1) for k = 2, . . . , n − 1. For ε > 0, sufficiently small, this new matrix satisfies all our requirements. The proof of the main claim of Theorem 2.16 is lengthy and complicated. We therefore divide it into a series of lemmas. We start with the following. Lemma 2.19 Let A = (aij ) be an m × m matrix with strictly positive entries. Assume aij ai+1,j+1 > c ai,j+1 ai+1,j


63

for all i, j = 1, . . . , m − 1. Then for k, ≥ 1 aij ai+k,j+ > ck ai,j+ ai+k,j , i = 1, . . . , m − k, j = 1, . . . , m − . Proof Multiply together the k inequalities ai+r,j+s ai+r+1,j+s+1 > c ai+r,j+s+1 ai+r+1,j+s for r = 0, 1, . . . , k − 1 and s = 0, 1, . . . , − 1. Lemma 2.20 The following are equivalent: (i) Fm (c) =

"m/2 m−j (−1)j c1j , j=0 j

m = 0, 1, 2, . . .

(ii) F0 (c) = F1 (c) = 1 Fm (c) = Fm−1 (c) − 1c Fm−2 (c),

m = 2, 3, . . .

where x denotes the integral part of x. It is a simple matter to verify the above lemma. Its proof is left to the reader. Lemma 2.21 Let Fm (c) be as above. Then (i) For c = 4 cos2 θ, Fm (c) = (ii) For ck = 4 cos2

sin(m + 1)θ . cm/2 sin θ

π k+1 ,

Fm−1 (ck ) −

1 1 Fm−2 (ck ) − m ≥ Fm (ck ) 2 ck ck

for k ≥ 3, m = 2, 3, . . . , k. Proof (i) follows from the trigonometric identity m/2 m − j sin(m + 1)θ (−1)j (2 cos θ)m−2j . = j sin θ j=0

To prove (ii), note that, from (ii) of Lemma 2.20, 1 1 1 1 1 − 2 Fm−2 (ck ) − m . Fm−1 (ck ) − 2 Fm−2 (ck ) − m − Fm (ck ) = ck ck ck ck ck

64


Now 4 cos2 θ =

sin 3θ + 1. sin θ

Thus continuing the above and using (i) we have (m−1)π

=

=

≥

1 ck − 1 sin k+1 − m c2k c m−2 c π 2 k sin k+1 k   (m−1)π 3π 1 1  sin k+1 sin k+1  − m−2  m+2  π π sin sin 2 π k+1 k+1 ck 2 cos k+1 3π sin (m−1)π sin k+1 1 k+1 − 1 ≥ 0, m+2 π π sin sin 2 k+1 k+1 c k

for k ≥ 3 and m = 2, 3, . . . , k. Looking at the last two terms we see that we actually have strict inequality except in the case k = 3 and m = 2. Note that from Lemma 2.21(i) we have that Fk (ck ) = 0 and Fk (cm ) > 0 for m < k. The proof of Theorem 2.16 is based on an induction argument. To ease exposition we assume, in the next series of lemmas, the following. Assumption A Let A = (aij ) be an m × m matrix with strictly positive entries such that aij ai+1,j+1 > c ai,j+1 ai+1,j for all i, j = 1, . . . , m − 1, r, . . . , m A > 0, r, . . . , m

r = 2, . . . , m,

(2.11)

(2.12)

and r + 1, . . . , m r + 2, . . . , m r, . . . , m − ar,r+1 ar+1,r A A > arr A r + 1, . . . , m r + 2, . . . , m r, . . . , m (2.13) for r = 1, . . . , m − 2. Lemma 2.22 Under Assumption A 1 r + 1, . . . , m r + 2, . . . , m r, . . . , m − ar+1,r+1 A > arr A A r + 1, . . . , m r + 2, . . . , m r, . . . , m c for r = 1, . . . , m − 2.


65

Proof Using (2.13) and (2.11) with i = j = r r + 1, . . . , m r + 2, . . . , m r, . . . , m − ar,r+1 ar+1,r A A > arr A r + 1, . . . , m r + 2, . . . , m r, . . . , m 1 r + 2, . . . , m r + 1, . . . , m > arr A − ar,r ar+1,r+1 A r + 2, . . . , m r + 1, . . . , m c 1 r + 2, . . . , m r + 1, . . . , m = arr A . − ar+1,r+1 A r + 2, . . . , m r + 1, . . . , m c

Lemma 2.23 Under Assumption A and if Fk (c) > 0, then 1 r, . . . , m r + 1, . . . , m − Fk−1 (c)arr A Fk (c)A r, . . . , m r + 1, . . . , m c 1 r + 1, . . . , m r + 2, . . . , m − Fk (c)ar+1,r+1 A > arr Fk+1 (c)A r + 1, . . . , m r + 2, . . . , m c for r = 1, . . . , m − 2. Proof As Fk (c) > 0 we multiply the result of Lemma 2.22 by Fk (c) to obtain r, . . . , m Fk (c)A r, . . . , m 1 r + 1, . . . , m r + 2, . . . , m > arr Fk (c)A − Fk (c)ar+1,r+1 A r + 1, . . . , m r + 2, . . . , m c for r = 1, . . . , m − 2. Thus 1 r, . . . , m r + 1, . . . , m − Fk−1 (c)ar,r A Fk (c)A r, . . . , m r + 1, . . . , m c 1 r + 1, . . . , m > arr Fk (c) − Fk−1 (c) A r + 1, . . . , m c r + 2, . . . , m 1 . − Fk (c)ar+1,r+1 A r + 2, . . . , m c Since Fk+1 (c) = Fk (c) −

1 Fk−1 (c) c

we have, continuing the above, 1 r + 1, . . . , m r + 2, . . . , m − Fk (c)ar+1,r+1 A . = arr Fk+1 (c)A r + 1, . . . , m r + 2, . . . , m c

66


Note the recursive nature of the inequality of Lemma 2.23. We use Lemma 2.23 in the proof of the next two lemmas. Lemma 2.24 Under Assumption A and if Fk+ (c) > 0 for = 0, 1, . . . , m− 1 − r, then 1 r, . . . , m r + 1, . . . , m − Fk−1 (c)ar,r A Fk (c)A r, . . . , m r + 1, . . . , m c > arr ar+1,r+1 · · · amm Fk+m−r+1 (c) for r = 1, . . . , m − 2. Proof We apply Lemma 2.23 m − 1 − r times, i.e., 1 r + 1, . . . , m r, . . . , m − Fk−1 (c)ar,r A Fk (c)A r + 1, . . . , m r, . . . , m c 1 r + 1, . . . , m r + 2, . . . , m > arr Fk+1 (c)A − Fk (c)ar+1,r+1 A r + 1, . . . , m r + 2, . . . , m c r + 2, . . . , m > arr ar+1,r+1 Fk+2 (c)A r + 2, . . . , m r + 3, . . . , m 1 − Fk+1 (c)ar+2,r+2 A r + 3, . . . , m c > ··· > arr ar+1,r+1 · · · am−2,m−2 1 m − 1, m − Fk+m−2−r (c)am−1,m−1 amm . × Fk+m−1−r (c)A m − 1, m c Now

A

m − 1, m m − 1, m

= am−1,m−1 amm − am−1,m am,m−1

and as am−1,m−1 amm > c am−1,m am,m−1 we have A

m − 1, m m − 1, m

1 am−1,m−1 amm − am−1,m−1 amm c 1 am−1,m−1 amm . = 1− c

>


67

Substituting in the above gives 1 r, . . . , m r + 1, . . . , m − Fk−1 (c)ar,r A Fk (c)A r, . . . , m r + 1, . . . , m c > arr ar+1,r+1 · · · am−2,m−2 1 Fk+m−1−r (c) 1 − am−1,m−1 amm c 1 − Fk+m−2−r (c)am−1,m−1 amm c 1 1 = arr ar+1,r+1 · · · amm Fk+m−1−r (c) 1 − − Fk+m−2−r (c) . c c From the recurrence relation of Lemma 2.20, Fk+m−r+1 (c)

Thus

1 Fk+m−r (c) − Fk+m−r−1 (c) c 1 1 = Fk+m−r−1 (c) − Fk+m−r−2 (c) − Fk+m−r−1 (c). c c =

Fk (c)A

1 r + 1, . . . , m r, . . . , m − Fk−1 (c)ar,r A r + 1, . . . , m r, . . . , m c > arr ar+1,r+1 · · · amm Fk+m−r+1 (c).

Lemma 2.25 Under Assumption A and if Fk (c) > 0, k = 1, . . . , s − 1, then 1, . . . , m A > 1, . . . , m 1 s + 1, . . . , m s + 2, . . . , m − Fs−1 (c)as+1,s+1 A a11 · · · ass Fs (c)A s + 1, . . . , m s + 2, . . . , m c for s = 1, . . . , m − 2. Proof We start with r = 1 in Lemma 2.22, which can be written as 1 1, . . . , m 2, . . . , m 3, . . . , m A > a11 A − a22 A 1, . . . , m 2, . . . , m 3, . . . , m c 1 2, . . . , m 3, . . . , m = a11 F1 (c)A − F0 (c)a22 A . 2, . . . , m 3, . . . , m c We now apply Lemma 2.23 s − 1 times for r = 2, . . . , s, with k = r − 1, to obtain our result.

68


We are now in a position to prove the sufficiency part of Theorem 2.16. The induction hypothesis will be that if A = (aij ) is an m × m (m ≥ 3) matrix with strictly positive entries, for which aij ai+1,j+1 > cm ai,j+1 ai+1,j ,

(2.14)

π ; then A is strictly totally for i, j = 1, . . . , m − 1, where cm = 4 cos2 m+1 positive and 3, . . . , m 2, . . . , m 1, . . . , m . − a12 a21 A A > a11 A 3, . . . , m 2, . . . , m 1, . . . , m

We first prove that this result holds in the case m = 3. It is easily verified that the inequality 1, 2, 3 2, 3 3 A > a11 A − a12 a21 A 1, 2, 3 2, 3 3 holds for every strictly totally positive matrix. Thus one only need verify (since c3 = 2) that if A is a 3 × 3 matrix with strictly positive entries, for which aij ai+1,j+1 > 2 ai,j+1 ai+1,j , for i, j = 1, 2, then A is strictly totally positive. That is, we must verify that det A > 0. Now 2, 3 2, 3 2, 3 − a12 A + a13 A det A = a11 A 2, 3 1, 3 1, 2 2, 3 2, 3 − a12 A . > a11 A 2, 3 1, 3 From (2.14) we have a11 a22 > 2 a12 a21 ,

a22 a33 > 2 a23 a32 .

Thus a11 a22 a33 > a12 a21 a33 + a11 a23 a32 , from which a11 A

2, 3 2, 3

= a11 a22 a33 − a11 a23 a32 > a12 a21 a33 2, 3 . > a12 a21 a33 − a12 a23 a31 = a12 A 1, 3

This proves the result in the case m = 3. We assume that our induction hypothesis is valid for m < n and prove it for m = n. We start with the following.


69

Proposition 2.26 Under the above assumptions 2, . . . , n 2, . . . , n − a1,j+1 A > 0, a1j A 1, . . . , j, . . . , n 1, . . . , j + 1, . . . , n j = 2, . . . , n − 1. Proof We use induction on the (n − 1) × (n − 1) submatrices 2, . . . , n 2, . . . , n A , A . 1, . . . , j, . . . , n 1, . . . , j + 1, . . . , n As cn−1 < cn we can apply the induction hypothesis, and thus both matrices are strictly totally positive and also satisfy (2.13) of Assumption A. We consider separately each of the two factors 2, . . . , n 2, . . . , n A , A . 1, . . . , j, . . . , n 1, . . . , j + 1, . . . , n By the Hadamard Inequality (Theorem 1.21) 2, . . . , n j + 2, . . . , n . A < a21 · · · aj+1,j A j + 2, . . . , n 1, . . . , j + 1, . . . , n Furthermore, from Lemma 2.19, a1j aj+1,j+1 > cjn a1,j+1 , aj+1,j . Thus

a1,j+1 A

2, . . . , n 1, . . . , j + 1, . . . , n

j + 2, . . . , n < a21 · · · aj+1,j a1,j+1 A j + 2, . . . , n 1 j + 2, . . . , n < j a1j a21 · · · aj,j−1 aj+1,j+1 A . j + 2, . . . , n cn Let us now consider

A

2, . . . , n 1, . . . , j, . . . , n

(2.15)

.

π As cn = 4 cos2 θn , θn = n+1 , and Fk (c) > 0 for c = 4 cos2 θ where θ < (see Lemma 2.21(i)), it follows that

Fk (cn ) > 0,

k = 1, . . . , n − 1.

π k+1

70


Thus we can apply Lemma 2.25 with s = j − 2 to obtain A

2, . . . , n j, j + 1, . . . , n > a21 · · · aj−1,j−2 Fj−2 (cn )A j − 1, j + 1, . . . , n 1, . . . , j, . . . , n 1 j + 1, . . . , n − Fj−3 (cn )aj,j−1 A . j + 1, . . . , n cn

By the induction hypothesis from (2.13) A

j, j + 1, . . . , n j − 1, j + 1, . . . , n j + 1, . . . , n j + 2, . . . , n − aj,j+1 aj+1,j−1 A , > aj,j−1 A j + 1, . . . , n j + 2, . . . , n

and thus A

2, . . . , n 1, . . . , j, . . . , n

j + 1, . . . , n > a21 · · · aj−1,j−2 aj,j−1 Fj−2 (cn )A j + 1, . . . , n j + 2, . . . , n −aj,j+1 aj+1,j−1 Fj−2 (cn )A j + 2, . . . , n j + 1, . . . , n 1 − Fj−3 (cn ) aj,j−1 A j + 1, . . . , n cn 1 j + 1, . . . , n = a21 · · · aj−1,j−2 aj,j−1 Fj−2 (cn ) − Fj−3 (cn ) A j + 1, . . . , n cn j + 2, . . . , n −aj,j+1 aj+1,j−1 Fj−2 (cn )A . j + 2, . . . , n

Since 1 Fj−1 (c) = Fj−2 (c) − Fj−3 (c) c and aj,j−1 aj+1,j+1 > c2n aj,j+1 aj+1,j−1


71

we have A

2, . . . , n 1, . . . , j, . . . , n

> a21 · · · aj−1,j−2

aj,j−1 Fj−1 (cn )A

j + 1, . . . , n j + 1, . . . , n

aj,j−1 aj+1,j+1 j + 2, . . . , n − Fj−2 (cn )A j + 2, . . . , n c2n j + 1, . . . , n = a21 · · · aj−1,j−2 aj,j−1 Fj−1 (cn )A j + 1, . . . , n aj+1,j+1 j + 2, . . . , n − . Fj−2 (cn )A j + 2, . . . , n c2n From Lemma 2.22 A

j + 1, . . . , n j + 1, . . . , n

1 j + 3, . . . , n j + 2, . . . , n . − > aj+1,j+1 A aj+2,j+2 A j + 3, . . . , n j + 2, . . . , n cn

Thus, finally, A

2, . . . , n 1, . . . , j, . . . , n

> a21 · · · aj−1,j−2 aj,j−1 aj+1,j+1

Fj−1 (cn )

j + 3, . . . , n 1 A − aj+2,j+2 A j + 3, . . . , n cn 1 j + 2, . . . , n − 2 Fj−2 (cn )A j + 2, . . . , n cn 1 = a21 · · · aj−1,j−2 aj,j−1 aj+1,j+1 Fj−1 (cn ) − 2 Fj−2 (cn ) cn j + 3, . . . , n j + 2, . . . , n 1 . − aj+2,j+2 Fj−1 (cn )A A j + 3, . . . , n j + 2, . . . , n cn

j + 2, . . . , n j + 2, . . . , n

(2.16)

72


We now combine (2.15) and (2.16) to obtain 2, . . . , n 2, . . . , n a1j A − a1,j+1 A 1, . . . , j, . . . , n 1, . . . , j + 1, . . . , n 1 > a1j a21 · · · aj−1,j−2 aj,j−1 aj+1,j+1 Fj−1 (cn ) − 2 Fj−2 (cn ) cn j + 3, . . . , n 1 j + 2, . . . , n − aj+2,j+2 Fj−1 (cn )A A j + 3, . . . , n j + 2, . . . , n cn 1 j + 2, . . . , n − j a1j a21 · · · aj,j−1 aj+1,j+1 A j + 2, . . . , n cn 1 1 = a1j a21 · · · aj−1,j−2 aj,j−1 aj+1,j+1 Fj−1 (cn ) − 2 Fj−2 (cn ) − j cn cn j + 3, . . . , n 1 j + 2, . . . , n . A − aj+2,j+2 Fj−1 (cn )A j + 3, . . . , n cn j + 2, . . . , n From Lemma 2.21(ii) Fj−1 (cn ) −

1 1 Fj−2 (cn ) − j ≥ Fj (cn ) c2n cn

for n ≥ 3, j = 2, 3, . . . , n. Since for these values of j and n we have Fj (cn ) ≥ 0, it follows that 2, . . . , n 2, . . . , n − a A a1j A 1,j+1 1, . . . , j, . . . , n 1, . . . , j + 1, . . . , n j + 2, . . . , n > a1j a21 · · · aj−1,j−2 aj,j−1 aj+1,j+1 Fj (cn )A j + 2, . . . , n 1 j + 3, . . . , n − aj+2,j+2 Fj−1 (cn )A . j + 3, . . . , n cn From Lemma 2.24 (with k = j and r = j + 2) the above quantity satisfies > a1j a21 · · · aj−1,j−2 aj,j−1 aj+1,j+1 aj+2,j+2 · · · ann Fn−1 (cn ). As Fn−1 (cn ) > 0 this proves Proposition 2.26. It now remains to advance the induction hypothesis, i.e., prove that A is strictly totally positive and (2.13) holds. This demands proving 1, . . . , n >0 (2.17) A 1, . . . , n

2.6 “Simple” criteria for strict total positivity and

A

1, . . . , n 1, . . . , n

> a11 A

2, . . . , n 2, . . . , n

− a12 a21 A

3, . . . , n 3, . . . , n

73

.

(2.18)

From Proposition 2.26 (which assumed the validity of the result for (n − 1) × (n − 1) matrices) n 2, . . . , n 1, . . . , n A = (−1)k+1 a1k A 1, . . . , n 1, . . . , k, . . . , n k=1 2, . . . , n 2, 3, . . . , n − a12 A . > a11 A 2, . . . , n 1, 3, . . . , n By the Hadamard Inequality 3, . . . , n 2, . . . , n A < a21 A 3, . . . , n 1, 3, . . . , n and thus

A

1, . . . , n 1, . . . , n

> a11 A

2, . . . , n 2, . . . , n

− a12 a21 A

3, . . . , n 3, . . . , n

.

This proves (2.18). As (2.12) holds for r = 2, . . . , n, by the induction hypothesis, and since we have now proven (2.18), which completes (2.13) for m = n (again applying the induction hypothesis), we have that the results of Lemmas 2.22–2.25 hold for m = n. From Lemma 2.25 (with m = n and s = n − 2) 1, . . . , n A 1, . . . , n 1 n − 1, n > a11 · · · an−2,n−2 Fn−2 (cn )A − Fn−3 (cn )an−1,n−1 ann . n − 1, n cn Now (see the proof of Lemma 2.24) 1 n − 1, n = an−1,n−1 ann − an−1,n an,n−1 > 1 − A an−1,n−1 ann . n − 1, n cn Thus

1 n − 1, n − Fn−3 (cn )an−1,n−1 ann n − 1, n cn 1 1 > Fn−2 (cn ) 1 − an−1,n−1 ann − Fn−3 (cn )an−1,n−1 ann cn cn 1 1 − Fn−3 (cn ) . = an−1,n−1 ann Fn−2 (cn ) 1 − cn cn

Fn−2 (cn )A

74


From the recurrence relation of Lemma 2.20(ii) 0 = Fn (cn )

1 Fn−2 (cn ) cn 1 1 = Fn−2 (cn ) − Fn−3 (cn ) − Fn−2 (cn ). cn cn =

Fn−1 (cn ) −

Substituting it follows that

A

1, . . . , n 1, . . . , n

> 0.

This advances the induction and completes the proof of Theorem 2.16.

2.7 Remarks Fekete’s Lemma 2.1 can be found in Fekete, P´ olya [1912]. (Different sections of this paper were written by the different authors.) The fact that for strictly totally positive matrices it suffices to consider only minors with consecutive rows and consecutive columns implies that the remaining minors are nonnegative combinations of the minors with consecutive rows and consecutive columns. If we detail the proof of Fekete’s Lemma 2.1 we see that the coefficients in this combination depend upon other minors of the same size, plus lower order minors, and in a nonlinear fashion. Such a formula is not valid for totally positive matrices in general. Theorem 2.3 was proved in Gasca, Pe˜ na [1992]. Their proof uses factorization, a subject to be considered in Chapter 6 (see also Metelmann [1973]). The two proofs given here are different. Proposition 2.5 first appears in Gasca, Pe˜ na [1992]. It is reproved in Gladwell [1998]. Theorem 2.6 is due to Whitney [1952]. Proposition 1.9 is also to be found in Whitney [1952]. Theorem 2.8 appears in Karlin [1968], p. 85. Proposition 2.9 can be found in Shapiro, Shapiro [1995]. Theorem 2.10 is in Cryer [1973]. Theorem 2.12 was conjectured in Cryer [1973] and proved in Cryer [1976] by very different methods. The proof given here can be found in Micchelli, Pinkus [1991]. The example of a 4 × 4 matrix, which is not totally positive but whose minors with consecutive rows and consecutive columns are nonnegative, is from Cryer [1973]. Theorem 2.13 was proved in Cryer [1976]. Theorem 2.14 was conjectured in Cryer [1973] and proved in Cryer [1976]. Proposition 2.15 is from Gasca, Pe˜ na [1993]. In Proposition 1.20, as a consequence of Theorem 1.19, we proved that a nonsingular totally positive matrix A is almost strictly totally positive if i + 1, . . . , i + p >0 A j + 1, . . . , j + p

2.7 Remarks

75

whenever ai+k,j+k > 0, k = 1, . . . , p, for all possible i, j and p. As a further consequence of Theorem 1.19, paralleling Proposition 2.5 and Proposition 2.9, it is not difficult to show that for a nonsingular totally positive matrix it is hardly necessary to verify the above conditions for all i, j, and p in order to determine if the matrix A is almost strictly totally positive. If we know the zero entries of A then, from Theorem 1.19, we can determine a minimal number of such conditions that must be verified. This is done using other methods in Gasca, Pe˜ na [1995], Gladwell [2004], and Gasca, Pe˜ na [2006]. Craven, Csordas [1998] first proved that if A = (aij ) is an n × m matrix with strictly positive entries and aij ai+1,j+1 ≥ c∗ ai,j+1 ai+1,j for all i = 1, . . . , n − 1, j = 1, . . . , m − 1, where c∗ = 4.07 . . . is the unique real root of x3 −5x2 +4x−1 = 0, then A is strictly totally positive. The fact that there is such a universal constant was in itself a surprising result. There was no claim made in the paper that their c∗ was best possible. On the other hand it is not difficult to show that we must have c∗ ≥ 4. Dimitrov, π Pe˜ na [2005] conjectured that c∗ = 4, in general, and cn = 4 cos2 n+1 is best possible for each n × n matrix. Unaware of Dimitrov, Pe˜ na [2005], this conjecture was proved by Katkova, Vishnyakova [2006]. It is their proof that is presented here. A simpler proof would be of interest.

3 Variation diminishing

The variation diminishing property of a matrix was introduced by I. J. Schoenberg in 1930. In Section 3.1 we prove two fundamental results concerning variation diminishing properties of totally positive matrices. The term “variation diminishing” was coined by P´ olya to describe the property whereby the number of sign changes of Ax is bounded above by the number of sign changes of x. The term is somewhat of a misnomer since it is not the “variation” that does not increase. The variation diminishing property and the exact form in which these bounds hold are the content of the two theorems of Section 3.1. In Section 3.2 we present one application. Other applications can be found in subsequent chapters.

3.1 Main equivalence theorems To explain the concept of variation diminishing we first fix some notation. We define two counts for the number of sign changes of a vector c = (c1 , . . . , cn ) ∈ Rn . These are S − (c) – the number of sign changes in the sequence c1 , . . . , cn , with zero terms discarded. Obviously S − (0) = 0. + S (c) – the maximum number of sign changes in the sequence c1 , . . . , cn , where zero terms are arbitrarily assigned values +1 or −1. For convenience we set S + (0) = n. Thus, for example, S − (1, 0, 2, −3, 0, 1) = 2, and S + (1, 0, 2, −3, 0, 1) = 4 . In our “S − count” we have a sign change from 2 to −3 and another from −3 to 1. In our “S + count,” by giving the first (left-most) 0 the value −1, 76

3.1 Main equivalence theorems

77

we get two additional sign changes. Giving the second 0 the value 1 or −1 does not alter the number of sign changes. The following elementary facts will be used. Their proof is left to the reader. Lemma 3.1 For every c = (c1 , c2 , . . . , cn ) ∈ Rn \{0} we have S + (c) + S − (! c) = n − 1, where ! c = (c1 , −c2 , . . . , (−1)n−1 cn ), i.e., we replace cr in c by (−1)r−1 cr for each r. Lemma 3.2 If lim ck = c

k→∞

then lim S − (ck ) ≥ S − (c)

k→∞

and lim S + (ck ) ≤ S + (c).

k→∞

The first main result is the following. Theorem 3.3 Let A be an n × m strictly totally positive matrix. Then (a) for each vector x ∈ Rm , x = 0, S + (Ax) ≤ S − (x); (b) if S + (Ax) = S − (x) and Ax = 0, then the sign of the first (and last) component of Ax (if zero, the sign given in determining S + (Ax)) agrees with the sign of the first (and last) nonzero component of x. Conversely, if (a) and (b) hold for some n×m matrix A and every x ∈ Rm , x = 0, then A is strictly totally positive. Proof We first prove that if A is strictly totally positive then (a) and (b) hold. Assume x = 0 and S − (x) = r. The components of x may be therefore divided into r + 1 blocks (x1 , . . . , xs1 ), (xs1 +1 , . . . , xs2 ), . . . , (xsr +1 , . . . , xm )

78

Variation diminishing

where, without loss of generality, we assume that xk (−1)i+1 ≥ 0,

k = si−1 + 1, . . . , si ,

for i = 1, . . . , r + 1, for each i strict inequality holds for at least one k, and where we set s0 = 0 and sr+1 = m. Let a1 , . . . , am ∈ Rn denote the column vectors of A and set si |xk |ak , i = 1, . . . , r + 1. yi = k=si−1 +1

Thus Ax =

m

xk ak =

r+1

(−1)i+1 yi .

i=1

k=1

Let Y be the n × (r + 1) matrix with columns y1 , . . . , yr+1 . Set w = Ax = Y d, where d = (1, −1, . . . , (−1)r ) ∈ Rr+1 . Now a simple calculation shows that s jp sj1 i1 , . . . , ip i1 , . . . , ip Y = ··· |xk1 | · · · |xkp |A . j1 , . . . , jp k1 , . . . , kp k1 =sj1 −1 +1

As

A

kp =sjp −1 +1

i1 , . . . , ip k1 , . . . , kp

>0

for all choices of 1 ≤ i1 < · · · < ip ≤ n, 1 ≤ k1 < · · · < kp ≤ m, and |xk1 | · · · |xkp | > 0 for some choice of admissible {k1 , . . . , kp } in the above sum, it follows that Y is an n × (r + 1) strictly totally positive matrix. We first prove that S + (Ax) ≤ S − (x). If n ≤ r + 1 there is nothing to prove. For if Ax = 0, then S + (Ax) ≤ n − 1 ≤ r = S − (x). If n = r + 1 then Ax = 0 since Ax = Y d where Y is a nonsingular (r + 1) × (r + 1) matrix and d = 0. Thus, if Ax = 0 then n ≤ r and S + (Ax) = n ≤ r = S − (x). We shall therefore assume that n > r + 1. If S + (Ax) ≥ r + 1, then there exist indices 1 ≤ i1 < · · · < ir+2 ≤ n such that εwij (−1)j+1 ≥ 0,

j = 1, . . . , r + 2,


79

for some ε ∈ {−1, 1} where w = Ax = Y d. Since Y is of rank r + 1, at least two of the wij are not zero. Consider the determinant wi1 .. .

y i1 1 .. .

···

wir+2

yir+2 1

···

yi1 r+1 .. .

.

yir+2 r+1

This determinant is zero since the first column is a linear combination of the other columns. Moreover expanding by this first column gives

r+2 i1 , . . . , ij , . . . , ir+2 j+1 0= (−1) wij Y . 1, . . . , r + 1 j=1 This equality cannot possibly hold since Y is strictly totally positive and wij (−1)j+1 is of one fixed (weak) sign and some of the wij are not zero. This contradiction implies that S + (Ax) ≤ r = S − (x). It remains to prove (b), namely that if S + (Ax) = S − (x) and Ax = 0, then the sign of the first (and last) component of Ax (if zero, the sign given in determining S + (Ax)) agrees with the sign of the first (and last) nonzero component of x. We continue our analysis using the above notation. We wish to prove that if S + (Ax) = S − (x) = r, Ax = 0, and εwij (−1)j+1 ≥ 0,

j = 1, . . . , r + 1,

where w = Y d = Ax, then ε = 1. Since every set of r + 1 rows of Y is linearly independent, we can solve for any component of d based on the values wi1 , . . . , wir+1 . Thus wi1 .. . 1=

wir+1

y i1 2 .. .

···

yi1 r+1 .. .

yir+1 2 · · · yir+1 r+1 i1 , . . . , ir+1 Y 1, . . . , r + 1

Expanding by the first column gives r+1

.

i1 , . . . , ij , . . . , ir+1 wij (−1)j+1 Y 2, . . . , r + 1 j=1 1= i1 , . . . , ir+1 Y 1, . . . , r + 1 As εwij (−1)j+1 ≥ 0,

j = 1, . . . , r + 1,

.

80


and Y is strictly totally positive, we obtain ε = 1. We now prove the converse direction. Its proof is surprisingly simple. We assume that S + (Ax) ≤ S − (x) for every x = 0, and if S + (Ax) = S − (x) and Ax = 0, then the sign of the first (and last) component of Ax (if zero, the sign given in determining S + (Ax)) agrees with the sign of the first (and last) nonzero component of x. We shall prove that A is a strictly totally positive matrix. Our proof is via induction on the size of the minors. We start as follows. Take ej , the jth unit vector in Rm . As S − (ej ) = 0 and its nonzero component is positive, we must have that Aej is a strictly positive vector. Thus the elements of A are all strictly positive. Now, assume that all (r − 1) × (r − 1) minors of A are strictly positive, 2 ≤ r ≤ min{n, m}, and consider the r × r submatrix i1 , . . . , ir . B=A j1 , . . . , jr

If

A

i1 , . . . , ir j1 , . . . , jr

=0

then there exists a z = (zj1 , . . . , zjr ) ∈ Rr \{0} such that Bz = 0. Let z ∈ Rm be the vector whose jk component is the above zjk , k = 1, . . . , r, and whose other components are zero. Then S − (z) ≤ r − 1 while S + (Az) ≥ r. This is a contradiction. Thus each r × r minor of A is nonsingular. We now let x = (xj1 , . . . , xjr ) ∈ Rr satisfy Bx = d where d = (1, −1, . . . , (−1)r+1 ). Let x ∈ Rm be the vector whose jk component is the above xjk , k = 1, . . . , r, and whose other components are zero. Note that S − (x) ≤ r − 1. From this construction of x we have (Ax)ik = (−1)k+1 , k = 1, . . . , r. Thus S + (Ax) ≥ r − 1. Since, by assumption, S − (Ax) ≤ S − (x) we necessarily have S − (Ax) = S − (x) = r − 1 and the sign patterns must also agree. Thus (−1)k+1 xjk > 0,

k = 1, . . . , r.


81

We now return to the equations Bx = d and solve for xj1 to obtain

r i1 , . . . , i , . . . , ir A j2 , . . . , jr =1 0 < xj1 = . i1 , . . . , ir A j1 , . . . , jr From the induction hypothesis the numerator is positive. Thus i1 , . . . , ir > 0. A j1 , . . . , jr For totally positive matrices the result is slightly weaker, but of the same form. Theorem 3.4 Let A be an n × m totally positive matrix. Then (a) for each vector x ∈ Rm , S − (Ax) ≤ S − (x); (b) if S − (Ax) = S − (x) and Ax = 0, then the sign of the first (and last) nonzero component of Ax agrees with the sign of the first (and last) nonzero component of x. Conversely, if (a) and (b) hold for some n×m matrix A and every x ∈ Rm , then A is totally positive. Proof Assume A is an n × m totally positive matrix. From Theorem 2.6 there exists a sequence of n × m strictly totally positive matrices (Ak ) for which lim Ak = A.

k→∞

Assume x = 0. Then from Theorem 3.3 S + (Ak x) ≤ S − (x). From Lemma 3.2 it follows that S − (Ax) ≤ lim S − (Ak x). k→∞

As S − (Ak x) ≤ S + (Ak x) always holds this implies S − (Ax) ≤ S − (x), which is (a).

82


If S − (Ax) = S − (x) = r and Ax = 0, then for all k sufficiently large we necessarily have, using Lemma 3.2 and Theorem 3.3, r = S − (Ax) ≤ S − (Ak x) ≤ S + (Ak x) ≤ S − (x) = r, and thus r = S − (Ax) = S − (Ak x) = S + (Ak x) = S − (x) = r. Since, for k sufficiently large, S − (Ak x) = S + (Ak x) the sign patterns of Ak x do not depend upon any zero entries. From Theorem 3.3 the sign patterns of Ak x agree with those of x. From a limiting argument these nonzero sign patterns of Ax and Ak x agree. This proves (b). The proof of the converse direction is essentially the same as the corresponding proof in Theorem 3.3. We assume that S − (Ax) ≤ S − (x) for every x, and if S − (Ax) = S − (x) then the sign of the first (and last) component of Ax agrees with the sign of the first (and last) nonzero component of x. We shall prove that A is a totally positive matrix. Our proof is by induction on the size of the minors. We start as follows. Take ej , the jth unit vector in Rm . As S − (ej ) = 0 and its nonzero component is positive we then have that Aej is a nonnegative vector. Thus the elements of A are all nonnegative. Now assume that all (r − 1) × (r − 1) minors are nonnegative, 2 ≤ r ≤ min{n, m}, and consider the r × r submatrix i1 , . . . , ir B=A . j1 , . . . , jr If B is singular, there is nothing to prove. Assume B is nonsingular. We now repeat the argument of the previous theorem. Let x = (xj1 , . . . , xjr ) ∈ Rr \{0} satisfy Bx = d where d = (1, −1, . . . , (−1)r+1 ). Let x ∈ Rm be the vector whose jk component is the above xjk , k = 1, . . . , r, and whose other components are zero. Note that S − (x) ≤ r − 1. From this construction of x we have (Ax)ik = (−1)k+1 , k = 1, . . . , r. Thus S − (Ax) ≥ r − 1. Since, by assumption, S − (Ax) ≤ S − (x) we necessarily have S − (Ax) = S − (x) = r − 1 and the sign patterns must also agree. Thus (−1)k+1 xjk > 0,

k = 1, . . . , r.

3.2 Intervals of strict total positivity

83

We now return to the equations Bx = d and solve for xj1 to obtain

r i1 , . . . , i , . . . , ir A j2 , . . . , jr =1 0 < xj1 = . i1 , . . . , ir A j1 , . . . , jr From the induction hypothesis the numerator is positive. Thus i1 , . . . , ir > 0. A j1 , . . . , jr We also have the following if A is both totally positive and nonsingular. Corollary 3.5 Let A be an n × n nonsingular totally positive matrix. Then for each x ∈ Rn S + (Ax) ≤ S + (x). Proof Let D denote the n × n diagonal matrix with diagonal entries alternately 1 and −1. From Proposition 1.6 DA−1 D is a nonsingular totally positive matrix. As such S − (DA−1 Dy) ≤ S − (y) for all y ∈ Rn . In addition, we recall from Lemma 3.1 that S + (c) + S − (Dc) = n − 1 for all c ∈ Rn \{0}. We may assume that x = 0, as there is nothing to prove if x = 0. Setting y = DAx we have x = A−1 Dy and from the above S + (Ax) = S + (Dy) = (n − 1) − S − (y) ≤ (n − 1) − S − (DA−1 Dy) = S + (A−1 Dy) = S + (x).

3.2 Intervals of strict total positivity As shall be seen in subsequent chapters, there are numerous applications of the variation diminishing property. We include here one such application. Let B = (bij ) and C = (cij ) be n × n matrices. We apply an ordering to these matrices. We say that B ≤∗ C if (−1)i+j bij ≤ (−1)i+j cij for all i, j. Equivalently, if D is the n × n diagonal matrix with diagonal entries

84


alternately 1 and −1, then DBD ≤ DCD where ≤ is the usual entrywise inequality. Theorem 3.6 Assume B = (bij ) and C = (cij ) are strictly totally positive n × n matrices satisfying B ≤∗ C. If A is an n × n matrix and B ≤∗ A ≤∗ C, then A is strictly totally positive. Proof We prove, under the above assumptions, that A is necessarily nonsingular. By continuity it then follows that the determinant of A is positive. Assume, for the moment, that we have proved this fact and let us see how it implies our theorem. Recall, from Theorem 2.2 (or Theorem 2.3), that to prove that A is strictly totally positive it suffices to show that i + 1, . . . , i + k A >0 j + 1, . . . , j + k for i = 0, . . . , n − k, j = 0, . . . , n − k, and k = 1, . . . , n. For each such i, j, and k consider the three k × k submatrices i + 1, . . . , i + k , B ij = B j + 1, . . . , j + k i + 1, . . . , i + k Aij = A , j + 1, . . . , j + k i + 1, . . . , i + k C ij = C . j + 1, . . . , j + k By assumption B ij ≤∗ Aij ≤∗ C ij for i + j even, while C ij ≤∗ Aij ≤∗ B ij for i + j odd. This implies, using an induction assumption, that i + 1, . . . , i + k > 0, A j + 1, . . . , j + k and the theorem is proved. Thus it suffices to prove the nonsingularity of A.

3.3 Remarks

85

We therefore assume that A is singular, and all k × k minors of A are strictly positive for k = 1, . . . , n − 1. Let x = 0 satisfy Ax = 0. As A is of rank n − 1 and 2, . . . , n >0 A 1, . . . , j, . . . , n for j = 1, . . . , n, it follows that j = 1, . . . , n − 1.

xj xj+1 < 0,

Assume, without loss of generality, that (−1)j xj > 0,

j = 1, . . . , n.

(3.1)

Since (−1)i+j bij ≤ (−1)i+j aij ,

i, j = 1, . . . , n,

we have i

(−1)

n

bij xj

=

j=1

n

i+j

(−1)

bij (−1) xj ≤ j

j=1

=

(−1)i

n

(−1)i+j aij (−1)j xj

j=1 n

aij xj = 0.

j=1

That is, (−1)i (Bx)i ≤ (−1)i (Ax)i = 0,

i = 1, . . . , n.

(3.2)

As B is nonsingular we have Bx = 0. Thus n − 1 = S + (Bx) = S − (x) = n − 1. However from (3.1) and (3.2) we see that the sign of the first (and last) component of Bx (if zero, the sign given in determining S + (Bx)) does not agree with the sign of the first (and last) nonzero component of x. This contradicts Theorem 3.3. Thus A is nonsingular.

3.3 Remarks According to Schoenberg [1930] the term variation diminishing (variationsvermindernd in German) was coined by P´ olya. To obtain the variation diminishing property of a matrix, i.e., S + (Ax) ≤ S − (x) or S − (Ax) ≤ S − (x) one does not need that the matrix A be strictly totally positive or totally positive, respectively. If A is an n × m matrix and n ≥ m then A satisfies S + (Ax) ≤ S − (x) for all x ∈ Rm \{0} if and only if A is

86


strictly sign regular (see e.g., Karlin [1968], p. 219, or Ando [1987], p. 29). An n × m matrix A is strictly sign regular if all its minors of order k are of one fixed sign εk ∈ {−1, 1} for each k = 1, . . . , m. If we permit minors to also vanish, then A is said to be sign regular. An n × m matrix A of rank m satisfies S − (Ax) ≤ S − (x) for all x ∈ Rm if and only if A is sign regular. This was first proven in Schoenberg [1930] with this rank condition (see also Motzkin [1936] and Schoenberg, Whitney [1951] for the more general case). The inequality n ≥ m and the rank condition, or something similar, is truly needed in these results. However, for strictly totally positive and totally positive matrices we have property (b) in the statements of Theorems 3.3 and 3.4. This property obviates the need for the rank condition and the inequality n ≥ m. This fact seems to be generally overlooked. In this regard see Brown, Johnstone, MacGibbon [1981], Lemma A.1, and Fan [1966], Theorems 5 and 6. For a generalization of sorts of variation diminishing, see Carnicer, Goodman, Pe˜ na [1995]. Garloff [1982] proved Theorem 3.6 by different methods. This theorem cannot hold for arbitrary totally positive matrices B and C. (Let B and C be matrices all of whose entries are 0 except for a single 1, suitably chosen.) In Garloff [1982] it is conjectured that the theorem holds if B and C are totally positive and nonsingular. In Garloff [1982] and Garloff [2003] the conjecture is proved for specific subclasses of such matrices.

4 Examples

In this chapter we present various examples of totally positive matrices. Some are specific examples, e.g., exponentials, powers, the Cauchy matrix, and the binomial coefficients. Others are examples of more general classes of matrices where we present conditions determining total positivity within the class, e.g., Green, Jacobi, Hankel, Toeplitz and Hurwitz matrices. Totally positive matrices are not closed under the operation of Hadamard products. In the last section we consider subclasses of totally positive matrices that are closed under this operation. We start, however, by discussing the connection between totally positive matrices, on the one hand, and totally positive kernels and Descartes systems, on the other hand.

4.1 Totally positive kernels and Descartes systems There are two types of continuous extensions of totally positive or strictly totally positive matrices or, from the opposite perspective, totally positive or strictly totally positive matrices are discrete counterparts of two continuous forms. A real-valued kernel K ∈ C(X × Y ) is said to be a totally positive or strictly totally positive kernel if n

(K(xi , yj ))i=1 m j=1 is, respectively, a totally positive or strictly totally positive matrix for every choice of x1 < · · · < xn in X, y1 < · · · < ym in Y and all possible n and m. If X and Y are finite sets then this is simply a definition of a totally positive or strictly totally positive matrix. As we shall see in the next two sections K(x, y) = exy is a strictly totally positive kernel on R × R, K(x, y) = xy 87

88

Examples

is a strictly totally positive kernel on R+ × R and K(x, y) = 1/(x + y) is a strictly totally positive kernel on R+ × R+ . The semicontinuous analogue of a strictly totally positive matrix, or semidiscrete analogue of a strictly totally positive kernel is given by a Descartes system. To define a Descartes system we start with a Chebyshev system. A Chebyshev system (generally abbreviated as a T -system) is a system of functions {u1 , . . . , ur } defined on a set X for which r

det (ui (xj ))i,j=1 = 0 for every choice of distinct x1 , . . . , xr in X. It is called a T+ -system if r

det (ui (xj ))i,j=1 > 0 for every choice of x1 < · · · < xr in X. A system of functions {u1 , . . . , ur } is said to be a Descartes system if every ordered subset of the u1 , . . . , ur is a T+ -system. Equivalently r

(ui (xj ))i=1 m j=1 is a strictly totally positive matrix for all x1 < · · · < xm in X. Descartes systems possess a variation diminishing property that follows easily from Theorem 3.3, often termed a Generalized Descartes Rule of Signs. It is that if {u1 , . . . , ur } is a Descartes system on an ordered set X, then for each nontrivial “polynomial” u(x) =

r

ai ui (x)

i=1

the number of zeros of u on X is bounded above by S − (a1 , . . . , ar ). As will be seen in the next section {xcj }rj=1 is a Descartes system on R+ for any c1 < · · · < cr .

4.2 Exponentials and powers Here are some elementary but instructive examples of strictly totally positive matrices. Certain specific cases have already been considered (e.g., see the proof of Theorem 2.6). Let A = (ebi cj )ni,j=1 where b1 < · · · < bn and c1 < · · · < cn . We claim the A is a strictly totally positive matrix. (In other words we are proving that K(x, y) = exy is a strictly totally positive kernel on R × R.) We present a two-step proof. We

4.2 Exponentials and powers

89

first prove that the determinant of A is nonzero, and then we prove that it is positive. Proof If A is singular there exists a (λ1 , . . . , λn ) ∈ Rn \{0} for which n

λj ebi cj = 0,

i = 1, . . . , n.

j=1

Set f (x) =

n

λj excj .

j=1

The function f is a nontrivial exponential polynomial with at least n distinct zeros b1 , . . . , bn . It is not identically zero since, for example, if m is the largest index for which λm = 0, then lim f (x)e−xcm = λm = 0.

x→∞

We claim that no such function exists, i.e., every exponential polynomial of this form has at most n − 1 distinct zeros, hence a contradiction, and A is nonsingular. We prove this claim by induction on n. It obviously holds for n = 1. Assume, as above, that f has at least n distinct zeros, n ≥ 2. Multiply f by e−xcn and differentiate to obtain g(x) = (f (x)e−xcn ) =

n−1

λj (cj − cn )ex(cj −cn ) .

j=1

The function g is an exponential polynomial with, by Rolle’s Theorem, at least n − 1 zeros. Thus, by our induction hypothesis, g is identically zero, i.e., λj = 0, j = 1, . . . , n − 1. Therefore f (x) = λn excn . As f has a zero we have λn = 0. Thus A is nonsingular. The (bi )ni=1 and (cj )nj=1 are arbitrary increasing sequences. As such, proving A nonsingular also proves the nonsingularity of all minors of A. Moreover, by continuity, the sign of the determinant of A is independent of the choice of the arbitrary increasing sequences (bi )ni=1 and (cj )nj=1 , but may depend upon n. We prove that it is positive for each and every n. Here are two proofs of the positivity of the determinant of A. Assume that all k × k determinants of the form of A are strictly positive for k = 1, . . . , n − 1. (The case k = 1 is obvious.) Let f (x) =

n j=1

λj excj

90

Examples

satisfy f (bi ) = 0, i = 1, . . . , n − 1 and f (bn ) = 1. Such an f exists and is unique by the nonsingularity of the associated matrix. In addition, n−1 det (ebi cj )i,j=1 λn = . det (ebi cj )ni,j=1

We proved earlier that f has at most n−1 zeros. Since f (bn ) = 1 it therefore follows that f is strictly positive for all x > bn−1 . As such λn = lim f (x)e−xcn > 0. x→∞

By our induction hypothesis n−1 det (ebi cj )i,j=1 >0

and therefore det (ebi cj )ni,j=1 > 0. A second proof uses the fact that the sign of the determinant of A is independent of the choice of the arbitrary increasing sequences (bi )ni=1 and (cj )nj=1 , but may depend upon n. As such it suffices to explicitly calculate the determinant of A for some specific choice of the (bi )ni=1 and (cj )nj=1 . Let cj = j − 1 and set ebi = xi . Then we have n−1 A = (xji )ni=1 j=0

where 0 < x1 < · · · < xn . This is the well-known Vandermonde matrix whose determinant is given by det A = (xk − x ) > 0. 1≤ 0) based on the following idea. Let G(x, y) =

m

λk xk y k ,

k=0

with λk ≥ 0, k = 0, 1, . . . , m. (We assume here that m is finite, but that

4.2 Exponentials and powers

91

need not be the case, and is not the case for G(x, y) = exy .) Then for any 0 < b1 < · · · < bn and 0 < c1 < · · · < cn the matrix A = (G(bi , cj ))ni,j=1 is totally positive. This follows easily from the fact    0 λ0 . . . 0 b1 . . . bm 1  ..   .. . . .  A =  ... . . . . ..   .  . b0n

. . . bm n

Set

0

B

b01  =  ... 

Λ

C

b0n

λ0  .. =  . 0  0 c1  .. =  . cm 1

c01 .. . cm 1

. . . λn



that  . . . c0n ..  . .. . .  m . . . cn

 . . . bm 1 ..  , .. . .  . . . bm n  ... 0 .  .. . ..  , . . . λn

 . . . c0n ..  . .. . .  m . . . cn

Thus A = BΛC. As B and C are strictly totally positive (they are Vandermonde matrices) and Λ is a diagonal totally positive matrix, the result follows. In this way it can be proven that ∞

G1 (x, y) = cosh xy =

x2k y 2k exy + e−xy = 2 (2k)! k=0

and ∞

x2k+1 y 2k+1 exy − e−xy = G2 (x, y) = sinh xy = 2 (2k + 1)! k=0

are strictly totally positive on R+ × R+ . These facts can also be proven directly as a consequence of the possible number of zeros of any linear combination of the exponentials exc1 , e−xc1 , . . . , excm , e−xcm and the parity of cosh xy and sinh xy.

92

Examples 4.3 Cauchy matrix

The matrix

A=

1 xi + yj

n i,j=1

is called a Cauchy matrix. For 0 < x1 < · · · < xn , 0 < y1 < · · · < yn it is strictly totally positive. We provide two proofs of this result. Proof I There is an explicit formula for the determinant of the above matrix and it is given by # # n 1 1≤ 0.

i=0

It is known that every polynomial of degree at most 2n that is nonnegative on [0, ∞) can be written in the form p(t) = (q(t))2 + t(r(t))2 where q and r are polynomials of degree at most n and n − 1, respectively. Let n n−1 q(t) = µj tj , r(t) = σ j tj . j=0

j=0

Then the condition 2n

ai bi > 0

i=0

whenever p(t) is a nonnegative (nontrivial) polynomial on [0, ∞) is easily shown to be equivalent to n

µj µk bj+k > 0

j,k=0

and n−1

σj σk bj+k+1 > 0

j,k=0

for every choice of (nontrivial) (µ0 , . . . , µn ) and (σ0 , . . . , σn−1 ). These latter conditions are exactly the strict positive definiteness of our two matrices. A minor digression is now in order. The study of moments was a central motivating theme in the development of functional analysis. An important precursor in this development is Stieltjes [1894–95]. In this paper Stieltjes discussed the problem of necessary and sufficient conditions on a sequence ∞ (bi )∞ i=0 so that the (bi )i=0 can be represented in the form , ∞ ti dα(t), i = 0, 1, 2, . . . bi = 0

for some nonnegative Borel measure dα. (We have rephrased Stieltjes’ original problem as there were then no Borel measures. In fact the Stieltjes

4.6 Hankel matrices

103

integral was introduced in that paper explicitly to deal with this problem.) We now know that this is essentially equivalent (via the Hahn–Banach Theorem) to asking for the existence of a continuous nonnegative linear functional L on C[0, ∞) satisfying L(ti ) = bi ,

i = 0, 1, 2, . . .

In the finite case things are somewhat simpler. We have , ∞ ti dα(t), i = 0, 1, . . . , 2n, bi = 0

(bi )2n i=0

(and the if and only if

are interior points of the associated convex moment cone) 2n

ai bi > 0

i=0

"2n i whenever p(t) = i=0 ai t is a nonnegative (nontrivial) polynomial on [0, ∞). It also follows from the theory of moments that each such (b0 , . . . , b2n ) can be represented in the form bi =

n+1

λj ξji ,

i = 0, 1, . . . , 2n,

(4.2)

j=1

where λj > 0, j = 1, . . . , n + 1, and 0 < ξ1 < · · · < ξn+1 . Let us therefore assume that the {bi } are as given in (4.2). Then, as is readily verified,   b0 . . . bn  .. . . ..  =  . . .  bn 

. . . b2n

ξ10  ..  . ξ1n Set

 0 λ1 . . . ξn+1 ..   .. .. . .  . n . . . ξn+1 0 

ξ10  .. X= . ξ1n

 ξ10 ... 0   . .. ..   ... . 0 ξn+1 . . . λn+1  0 . . . ξn+1 ..  .. . .  n . . . ξn+1

 . . . ξ1n ..  . .. . .  n . . . ξn+1

104

Examples

and



λ1  .. Λ= . 0

 ... 0 ..  . .. . .  . . . λn+1

Thus A = XΛX T . As X and X T are strictly totally positive matrices (X is a Vandermonde matrix) and Λ is a diagonal nonsingular totally positive matrix, the result follows. That is, i1 , . . . , ir A = j1 , . . . , jr 1≤k1 0, an = 0 and ak = 0 for k < 0 and k > n. Then A = (aj−i )∞ i,j=1 is a totally positive matrix if and only if p(x) =

n

ak xk

k=0

has n negative zeros. Furthermore, if A is totally positive then i1 , . . . , ir >0 A j1 , . . . , jr if and only if ajk −ik > 0,

k = 1, . . . , r,

i.e., 0 ≤ jk − ik ≤ n, k = 1, . . . , r. Proof Let us first assume that p has n negative zeros. Then, assuming a0 = 1, we have n p(x) = (1 + αr x) r=1

where αr > 0 for r = 1, . . . , n. Let (r)

Er = (bj−i )∞ i,j=1 (r)

(r)

(r)

where b0 = 1, b1 = αr and bk = 0 for k = 0, 1. It follows by inspection, since αr > 0, that each Er is totally positive. Furthermore A=

n

Er .

r=1

This latter result follows from the fact that the generating function of a

106

Examples

product of Toeplitz matrices is the product of their generating functions. As A is a product of totally positive matrices, then A is totally positive. Let us now assume that A is totally positive. We will prove that p(x) =

n

ak xk

k=0

has n negative zeros. We prove this by contradiction using the variation diminishing property of totally positive matrices. From the fact that A is totally positive (thus ak ≥ 0) and a0 > 0, it follows that p has no nonnegative real zeros. Let us assume that p has nonreal zeros. Since the coefficients of p are real, if p(w) = 0 then p(w) = 0. Thus we may assume that w is a zero of p satisfying w = reiθ where 0 < θ < π. Now, as n

∈Z

ak wk+ = 0,

k=0

we have

n

∈ Z.

ak rk+ sin(k + )θ = 0,

k=0

Let x = (x0 , x1 , x2 , . . .) where xk = rk sin kθ. Thus Ax = 0. Let A[m] denote the m × (n + m) submatrix of A obtained by choosing the first m rows and (n + m) columns of A, i.e.,   a0 · · · an 0   .. .. A[m] =  . . . 0

a0

···

an

Set x[m] = (x0 , . . . , xn+m−1 ). Thus A[m] x[m] = 0 ∈ Rm .

4.7 Toeplitz matrices

107

As n ≥ 1 and because rank A[m] = m there exists an x∗ ∈ Rn+m satisfying A[m] x∗ = d where d = (1, −1, , . . . , (−1)m−1 ) ∈ Rm . Thus for any ε A[m] (x[m] + εx∗ ) = εd. From the variation diminishing properties of the totally positive matrix A[m] (Theorem 3.4) we have m − 1 = S − (εd) ≤ S − (x[m] + εx∗ ) for any ε = 0. Take ε∗ = 0 and sufficiently small such that (x[m] + εx∗ )k (x[m] )k > 0 if (x[m] )k = xk = 0. Note that if xk = 0, 0 < k < n + m − 1, then since 0 < θ < π we have xk−1 xk+1 < 0. Thus S − (x[m] + εx∗ ) ≤ S − (x[m] ) + 2 (because we do have x0 = 0 and can have xn+m−1 = 0). Therefore m − 1 ≤ S − (x[m] ) + 2, i.e., m − 3 ≤ S − (x[m] ) for all m. Now, as is easily seen, (n + m)θ π simply because sin α > 0 for 0 < α < π and sin α < 0 for π < α < 2π, and the fact that when going from xk to xk+1 we alter the “argument” of xk by θ. Thus (n + m)θ m−3≤ π S − (x[m] ) ≤

108

Examples

for every m. But θ < 1, π which implies that the above cannot hold for m sufficiently large. We have arrived at a contradiction. The last claim of the theorem (namely that A is necessarily an almost strictly totally positive matrix) is fairly elementary. One direction is a consequence of Theorem 1.13, i.e., i1 , . . . , ir A >0 j1 , . . . , jr implies that the diagonal elements of this minor are strictly positive. The converse direction is a simple consequence of Theorem 1.19 and the fact that i + 1, . . . , i + r A = ar0 > 0 i + 1, . . . , i + r and

A

i + 1, . . . , i + r i + 1 + n, . . . , i + r + n

= arn > 0.

The above characterization does not necessarily mean that it is a simple matter to determine whether a given Toeplitz matrix is totally positive or not. It is simply equivalent to the fact that the associated generating polynomial has all negative zeros. We return to this problem in Section 4.9. Let us consider a few specific examples of totally positive Toeplitz matrices and totally positive matrices derived from Toeplitz matrices. 1. Consider the polynomial p(x) = (1 + x)n . As its zeros are all negative the Toeplitz matrix with n , k = 0, 1, . . . , n ak = k is totally positive. 2. In Theorem 1.11 we proved that if the square matrix A is (strictly) totally positive, then the matrix B defined by b1j = a1j ,

for all j,

4.7 Toeplitz matrices and for i ≥ 2 bij =

bi−1,k akj ,

109

for all j,

k

is also (strictly) totally positive. Applying this result to the above Toeplitz matrix where n = 1, i.e., a0 = a1 = 1, am = 0 if m = 0, 1, we get b11 = b12 = 1 and for i ≥ 2 bi1 = 1,

bij = bi−1,j−1 + bi−1,j ,

j ≥ 2,

which immediately implies that i , i, j = 1, 2, . . . . bij = j−1 n Letting m = 0 for n < m we have that the matrix of binomial coefficients ∞ i C= j i,j=0 is totally positive. This is equivalent to the fact that the infinite Toeplitz matrix ∞ 1 D= (i − j)! i,j=0 is totally positive. 3. Applying Theorem 1.11 to the infinite totally positive Toeplitz matrix A = (aij )∞ i,j=0 , where 1, i ≤ j aij = 0, i > j it easily follows that the matrix ∞ i+j B= j i,j=0 is totally positive, which is equivalent to the fact that the Hankel matrix ∞

D = ((i + j)!)i,j=0 is totally positive. The total positivity of D can also be proven by other methods.

110

Examples

4. Consider the polynomial p(x) =

n

(1 + q i−1 x)

i=1

for some q > 0, q = 1. These polynomials are called Gaussian polynomials. Set n p(x) = ak xk k=0

and let us calculate the coefficients ak . From the fact that (1 + q n x)p(x) = (1 + x)p(qx) we have (1 + q n x)

n

ak xk = (1 + x)

k=0

n

ak q k xk .

k=0 k

Equating the coefficients of x we obtain for k = 1, . . . , n ak + q n ak−1 = ak q k + ak−1 q k−1 implying ak = ak−1 q k−1

(1 − q n−k+1 ) . (1 − q k )

Now a0 = 1 so that ak = q (k−1)+(k−2)+···+1 which we write as ak = The

-n. k

= q

(1 − q n−k+1 ) · · · (1 − q n ) , (1 − q k ) · · · (1 − q)

-n. k

q k(k−1)/2 . q

(1 − q n−k+1 ) · · · (1 − q n ) (1 − q k ) · · · (1 − q)

are termed q-binomial coefficients. Thus the Toeplitz matrix with -n. q k(k−1)/2 , ak = k q is totally positive.

k = 0, 1, . . . , n,

4.8 Generalized Hurwitz matrices

111

4.8 Generalized Hurwitz matrices Assume we are given real numbers a0 , a1 , . . . , an where n ≥ M ≥ 2. For ease of exposition, set aj = 0 for j < 0 and j > n. Let us define P = (pij )∞ i,j=1 where pij = aM j−i , Thus

          P =         

i, j = 1, 2, . . . .

aM −1 aM −2 .. .

a2M −1 a2M −2 .. .

a3M −1 a3M −2 .. .

a0 0 0 .. .

aM aM −1 aM −2 .. .

a2M a2M −1 a2M −2 .. .

0 0 .. .

a0 0 .. .

aM aM −1 .. .

 ··· ···      ···    ···   ··· .     ···   ···  

The matrix P is not a Toeplitz matrix, but it resembles a Toeplitz matrix. It also contains M Toeplitz submatrices. Simply take all columns and every M th row. Somewhat surprisingly we prove that there are a finite number of determinantal conditions that imply that this matrix is totally positive. Theorem 4.6 Assume a0 > 0 and k, k + 1, . . . , k + r − 1 >0 P 1, 2, . . . , r

(4.3)

for k = 1, . . . , M − 1, and r = 1, . . . , n+k−1 M −1 . Then P is totally positive, ak > 0, k = 0, 1, . . . , n, and i1 , . . . , ir >0 P j1 , . . . , jr if and only if pik jk = aM jk −ik > 0, i.e., 0 ≤ M jk − ik ≤ n.

k = 1, . . . , r,

112

Examples

Before we prove this theorem note that for r > n+k−1 M −1 the last column of

k, k + 1, . . . , k + r − 1 P 1, 2, . . . , r

vanishes identically. That is, the conditions of (4.3), together with a0 > 0, are exactly the demand that all possible nonvanishing minors composed from consecutive rows and initial consecutive columns be strictly positive. In addition, let k ∈ {1, . . . , M − 1} be such that r=

n+k−1 M −1

is an integer. Such a k exists and r ≥ 2. For this k and r k, . . . , k + r − 2 k, . . . , k + r − 1 . = an P P 1, . . . , r − 1 1, . . . , r Thus, from (4.3), it immediately follows that an > 0. Our proof is via induction on n. In this proof we use the following result. Lemma 4.7 Let k = 1, . . . , M − 1, j = 0, 1, . . . ,

bM j+k−1 = aM j+k , and bM j−1 = aM j −

a0 aM j+1 , a1

j = 1, 2, . . . ,

where applicable, to define b0 , . . . , bn−1 . Set bk = 0 for k < 0 and k > n − 1. Define Q = (qij )∞ i,j=1 where qij = bM j−i ,

i, j = 1, 2, . . . .

Then Q satisfies the conditions of Theorem 4.6 (with n − 1 replacing n). Proof Note that b0 = a1 = pM −1,1 > 0 and by the above b−1 = a0 −

a0 a1 = 0. a1

Thus bj = 0 for j < 0 and, by definition, bj = 0 for j > n − 1.

4.8 Generalized Hurwitz matrices

113

Recall that           P =         

aM −1 aM −2 .. .

a2M −1 a2M −2 .. .

a3M −1 a3M −2 .. .

a0 0 0 .. .

aM aM −1 aM −2 .. .

a2M a2M −1 a2M −2 .. .

0 0 .. .

a0 0 .. .

aM aM −1 .. .

 ··· ···      ···    ···   ···      ···   ···  

and Q has exactly this same form, i.e.,           Q=         

bM −1 bM −2 .. .

b2M −1 b2M −2 .. .

b3M −1 b3M −2 .. .

b0 0 0 .. .

bM

b2M

bM −1 bM −2 .. .

b2M −1 b2M −2 .. .

0 0 .. .

b0 0 .. .

bM bM −1 .. .

 ··· ···      ···    ···   ··· .     ···   ···  

For k = 2, . . . , M − 1, it is easily verified that Q

k, k + 1, . . . , k + r − 1 1, 2, . . . , r

=P

k − 1, k, . . . , k + r − 2 1, 2, . . . , r

>0

= (n−1)+k−1 . We must also consider for r = 1, . . . , n+(k−1)−1 M −1 M −1 Q

1, 2, . . . , r 1, 2, . . . , r

.

114

Examples

−2 Now, for 2 ≤ r ≤ n+M M −1 ,

0

0 we have

Q

aM +1 aM aM −1 .. .

1, 2, . . . , r − 1 1, 2, . . . , r − 1

a2m+1 a2M a2M −1 .. .

a2M +1 a2M −1 .. .

··· ··· .. .

··· ··· .. .

>0

n−1 for 1 ≤ r − 1 ≤ M −1 , which is exactly what we wished to prove.

We now proceed to the proof of our theorem. Proof of Theorem 4.6 We start with the case  0 aM −1  aM −2 0   . .. ..  .   a 0 1   aM  a0 P =  0 aM −1  .. ..   . .   0 a 0   0 0  .. .. . .

n = M . Here  ··· ···      ···    ···   ··· .  ..  .   ···   ···  

From (4.3) we have pk1 = aM −k > 0, k = 1, . . . , M − 1, and M − 1, M a1 0 0
0. The other claims follow easily from this fact and the geometric form of P . We now apply induction assuming the result is valid for n − 1, where n > M . We start with the a0 , . . . , an and P as in the statement of Theorem 4.6. From Lemma 4.7, the b0 , . . . , bn−1 and the Q therein satisfy (4.3) for n − 1, and therefore Q is totally positive, bk > 0, k = 0, . . . , n − 1, and i1 , . . . , ir >0 Q j1 , . . . , jr if and only if qik jk = bM jk −ik > 0,

k = 1, . . . , r,

i.e., 0 ≤ M jk − ik ≤ n − 1. From Lemma 4.7 we have aM j+k aM j

= bM j+k−1 ,

k = 1, . . . , M − 1

= bM j−1 + cbM j ,

k = 0,

(4.4)

where c > 0. (Specifically c = a0 /a1 and we are given a0 > 0 and a1 = pM −1,1 > 0.) Thus P is obtained from Q by an addition of a positive multiple of one row to a succeeding row and shifting (renumbering) of the rows. Both these operations (see Proposition 1.5) preserve totally positive. Thus P is totally positive. The fact that a0 , a1 , . . . , an > 0 also follows easily. We are given a0 > 0, and the positivity of the other ak is a consequence of the positivity of b1 , b1 , . . . , bn−1 and (4.4). For any nonsingular totally positive matrix the diagonal elements are positive (see Theorem 1.13). That is, if i1 , . . . , ir P >0 j1 , . . . , jr then pik jk = aM jk −ik > 0,

k = 1, . . . , r,

i.e., 0 ≤ M jk − ik ≤ n. It remains to prove the converse. As such let us assume that 0 ≤ M jk − ik ≤ n, We wish to prove that

P

i1 , . . . , ir j1 , . . . , jr

k = 1, . . . , r. > 0.

116

Examples

Based on (4.4) we can write pis jt = aM jt −is = bM jt −is −1 + c(is )bM jt −is where

c(is ) =

Now

P

i1 , . . . , ir j1 , . . . , jr

0, is is not a multiple of M c, is is a multiple of M .

= det (pis jt ) = det (bM jt −is −1 + c(is )bM jt −is ).

Thus the desired minor of P is the nonnegative sum of various minors of Q. All the minors of Q are nonnegative. Our task is to show that at least one of the minors in this sum is positive, and has positive coefficient. This we do by defining is = is + 1 if 0 < M js − is ≤ n and is = is (Note that in the latter case As

if M js − is = 0.

c(is )

= c(is ) = c > 0.)

0 ≤ M jk − ik ≤ n,

k = 1, . . . , r,

then from the above construction 0 ≤ M jk − ik ≤ n − 1,

k = 1, . . . , r.

Furthermore i1 < · · · < ir . To see this note that if is = is + 1 then obviously is−1 < is , while if is = is then M js = is and is−1 ≤ M js−1 < M js = is . Therefore

Q

i1 , . . . , ir j1 , . . . , jr

> 0.

This minor, with positive coefficient, is one of the minors involved in calculating the desired minor of P . Thus i1 , . . . , ir > 0, P j1 , . . . , jr which proves the theorem.

4.9 More on Toeplitz matrices Remark The matrix

     P =    

a1 a0 0 0 0 .. .

a3 a2 a1 a0 0 .. .

a5 a4 a3 a2 a1 .. .

··· ··· ··· ··· ··· .. .

117

         

is called a Hurwitz matrix. This is the matrix of Theorem 4.6 with M = 2. The conditions (4.3) of Theorem 4.6 in this case reduce to 1, . . . , r > 0, r = 1, . . . , n. P 1, . . . , r These inequalities, together with a0 > 0, are equivalent to the fact that the polynomial p(x) = a0 xn + a1 xn−1 + · · · + an is a Hurwitz polynomial, i.e., has all its zeros in the open left-hand plane Re (z) < 0. If p has all its zeros in the closed left-hand plane Re (z) ≤ 0, then P is totally positive. Unfortunately the converse is not true.

4.9 More on Toeplitz matrices In Section 4.7 we considered infinite Toeplitz matrices with a finite number of nonzero terms and proved that the matrix is totally positive if and only if its associated generating polynomial has only negative zeros. In other words, we replaced one problem by another one which, while providing an insight into the theory, does not always constitute a useful criterion for determining whether a given matrix is in fact totally positive. In this short section we apply the theory of Hurwitz matrices in order to generate verifiable conditions for when such a matrix is totally positive, i.e., when a polynomial has only negative zeros. We present most of the results here without proof. Their proofs would take us too far afield. References can be found in the last section of this chapter. Let n ak xn−k p(x) = k=0

118

Examples

with a0 > 0, and set q(x) = p(x2 ) + xp (x2 ). Then we have Theorem 4.8 The polynomial q is a Hurwitz polynomial if and only if p has simple negative zeros. Let q(x) =

2n

bk x2n−k .

k=0

Then b2k = ak ,

k = 0, 1, . . . , n,

and b2k+1 = (n − k)ak ,

k = 1, . . . , n − 1.

Let A be the infinite Toeplitz matrix based on p, i.e.,  a0 a1 a2 · · · an 0 0  0 a0 a1 · · · an−1 a 0 n  A =  0 0 a ··· a a a 0 n−2 n−1 n  .. .. .. . . .. .. .. . . . . . . . and B be the  b1  b  0   0 B=  0   0  .. .

··· ··· ··· .. .

    

infinite Hurwitz matrix based on q, i.e.,   b 3 b5 · · · na0 (n − 1)a1 (n − 2)a2  a b 2 b4 · · ·  a1 a2   0   b1 b3 · · ·   0 na0 (n − 1)a1   b0 b2 · · ·  =  0 a0 a1   0 b1 · · ·   0 0 na0   .. .. .. .. . . .. . . . . . .

··· ··· ··· ··· ··· .. .

     .    

Theorem 4.9 The following are equivalent: (i) A is totally positive. (ii) B is totally positive. (iii) p has n negative zeros. Proof From Theorem 4.5, conditions (i) and (iii) are equivalent. Furthermore, if B is totally positive, then since A is a submatrix of B it follows that A is totally positive.

4.10 Hadamard products of totally positive matrices

119

Assume p has n negative zeros. Perturb these zeros so that they are simple and negative. Then from Theorem 4.8 q is a Hurwitz polynomial and thus the associated Hurwitz matrix B is totally positive. Perturb back. From the results of Section 4.8, b0 = a0 > 0 and 1, . . . , k > 0, k = 1, . . . , 2n, B 1, . . . , k is equivalent to the fact that q is a Hurwitz polynomial. Thus we have the following. Theorem 4.10 The polynomial p(x) =

n

ak xn−k

k=0

with a0 > 0 has n simple negative zeros if and only if 1, . . . , k > 0, k = 1, . . . , 2n. B 1, . . . , k When considering the total positivity of the associated Toeplitz matrix, the necessity of ak > 0,

k = 0, 1, . . . , n,

is obvious. With this a priori assumption it suffices to verify the positivity of half of the above determinants. Proposition 4.11 Let p be as above and assume that ak > 0, k = 0, 1, . . . , n. Then p has n simple negative zeros if and only if 1, . . . , 2k + 1 > 0, k = 1, . . . , n − 1. B 1, . . . , 2k + 1 When p has multiple zeros then q has zeros on the imaginary axis and the situation is more complicated.

4.10 Hadamard products of totally positive matrices Given n × m matrices A = (aij ) and B = (bij ) we define their Hadamard product as the n × m matrix formed by the entrywise product of the elements of A and B, i.e., it is the n × m matrix C = (cij ) where cij = aij bij .

120

Examples

We use the notation C = A ◦ B. While certain classes of matrices are Hadamard products, the class of totally them. For example, the matrix  1 1  A= 1 1 0 1

closed under the operation of positive matrices is not one of  1 1  1

is totally positive, while  1 1 0 A ◦ AT =  1 1 1  0 1 1 

has determinant equal to −1. (As strictly totally positive matrices are dense in the class of totally positive matrices, it follows that there also exist strictly totally positive matrices whose Hadamard product is not totally positive.) However, there are numerous subclasses of totally positive matrices that are closed under the operation of taking Hadamard products. In this section we detail some of them. 1. Consider a strictly totally positive matrix satisfying the criteria in Section 2.6. That is, let A = (aij ) be an n × n matrix, all of whose entries are strictly positive and where π 2 ai,j+1 ai+1,j aij ai+1,j+1 > 4 cos n+1 for all i, j = 1, . . . , n − 1. Now let B = (bij ) be any n × n matrix, all of whose entries are strictly positive, and such that i, i + 1 B ≥0 j, j + 1 for all i, j = 1, . . . , n − 1. Then from Theorem 2.16 the Hadamard product A ◦ B = (aij bij ) is strictly totally positive.


121

2. Green’s matrices. If A and B are totally positive Green’s matrices, then A◦B is also a totally positive Green’s matrix. This immediately follows from the definition of a Green’s matrix and the characterization of Green matrices as given in Theorem 4.2. 3. Jacobi matrices. We recall that a Jacobi matrix is a square tridiagonal matrix, i.e., a matrix A = (aij )ni,j=1 where aij = 0 if |i − j| ≥ 2. Furthermore, from Theorem 4.3, a Jacobi matrix is totally positive if and only if all its off-diagonal elements and all its principal minors containing consecutive rows and columns are nonnegative. We prove the following. Proposition 4.12 The Hadamard product of two totally positive Jacobi matrices is a totally positive Jacobi matrix. Proof Assume we are given the  a11  a21   A= 0  .  .. 0

totally positive Jacobi matrix  a12 0 · · · 0 a22 a23 · · · 0   a32 a33 · · · 0  . .. .. ..  .. . . . .  0 0 · · · ann

Let c ≥ d ≥ 0, c > 0. Then, assuming a11 > 0, the matrix  a12 0 ··· 0 ca11  0 12 a21 a22 − daca a · · · 0 23  11  a33 · · · 0 0 a32 Ac,d =   . .. .. .. ..  . .  . . . . 0 0 0 · · · ann is also a totally positive Jacobi matrix. To  ca11 a12 0  da21 a22 a23  a32 a33 !c,d =  A  0  . .. ..  .. . . 0

0

0

       

see this let ··· ··· ··· .. .

0 0 0 .. .

···

ann

    .  

!c,d is totally positive since it is obtained from A by multiplying The matrix A the first column of A by d and adding (c − d)a11 to the (1, 1) entry. Both

122

Examples

operations preserve total positivity (see Proposition 1.5). Now consider Ac,d . Its off-diagonal elements are nonnegative and i, . . . , j i, . . . , j ! = Ac,d Ac,d i, . . . , j i, . . . , j except when i = 2, in which case 1 ! 2, . . . , j 1, . . . , j Ac,d = ≥ 0. Ac,d 2, . . . , j 1, . . . , j ca11 Thus Ac,d is a totally positive Jacobi matrix. Let us now assume that A and B are n × n totally positive Jacobi matrices. Our proof is by induction on n. The result is easily verified for n = 1 (and n = 2). Assume the result holds for r × r matrices, r ≤ n − 1. Now   a11 b11 a12 b12 0 ··· 0  a21 b21 a22 b22 a23 b23 · · ·  0     0 a32 b32 a33 b33 · · · 0 A◦B = .   .. .. .. .. ..   . . . . . 0

0

We wish to prove that

(A ◦ B)

···

0

1, . . . , n 1, . . . , n

ann bnn

≥0

to advance the induction. If a11 = 0 then a12 = 0 or a21 = 0, and the above determinant is therefore zero. The similar result holds if b11 = 0. We may b21 therefore assume that a11 , b11 > 0. Multiply the first row of A ◦ B by aa21 11 b11 and subtract it from the second row to obtain   a12 b12 0 ··· 0 a11 b11   a21 b21 a23 b23 · · · 0 0 a22 b22 − a12ab12   11 b11   b a b · · · 0 0 a .  32 32 33 33 D=  .. .. .. .. ..   .   . . . . 0 Therefore

0

(A ◦ B)

1, . . . , n 1, . . . , n

0

= a11 b11 D

··· 2, . . . , n 2, . . . , n

ann bnn .

We now apply the induction hypothesis to the (n−1)×(n−1) submatrix of D obtained by deleting its first row and column. This matrix is the


123

Hadamard product of the two (n − 1) × (n − 1) totally positive Jacobi matrices 2, . . . , n 2, . . . , n . and B Ab11 b22 ,b12 b21 2, . . . , n 2, . . . , n The former matrix is totally positive by our previous analysis and since b11 b22 ≥ b12 b21 . This proves the proposition. Remark The same proof shows that the Hadamard product of a totally positive Jacobi matrix and an arbitrary totally positive matrix is a totally positive Jacobi matrix. 4. Hankel matrices. We are given Hankel matrices A = (ai+j )ni,j=0 and B = (bi+j )ni,j=0 . If A and B are strictly totally positive, then A ◦ B is strictly totally positive. This follows from the fact that A ◦ B = (ai+j bi+j )ni,j=0 is also a Hankel matrix, the characterization of strictly totally positive Hankel matrices as detailed in Theorem 4.4 and the Schur Product Theorem which states that if C and D are two strictly positive definite matrices, then so is their Hadamard product C ◦ D. 5. Toeplitz matrices. In Theorem 4.5 we considered Toeplitz matrices of the form A = (aj−i )∞ i,j=1 where a0 > 0, an = 0 and ak = 0 for k < 0 and k > n, and showed that A is totally positive if and only if p(x) =

n

ak xk

k=0

has n negative zeros. A special case of Maló’s Theorem states that if p(x) =

n

ak xk

k=0

has n negative zeros and q(x) =

m k=0

bk xk

124

Examples

has m negative zeros, then the polynomial r

ak bk xk

k=0

has r negative zeros, where r = min{n, m}. Thus the Hadamard product of two totally positive Toeplitz matrices of the above form is again a totally positive Toeplitz matrix. 6. Hurwitz matrices. As stated in the remark at the end of Section 4.8, the polynomial n p(x) = ak xk k=0

is a Hurwitz polynomial if and only if all its zeros lie in the open left-hand plane Re (z) < 0. Assuming a0 > 0, we have that p is a Hurwitz polynomial if and only if the matrix   a1 a3 a5 · · ·  a a a ···  2 4  0     0 a1 a3 · · ·   P =  0 a a ···   0 2    0 0 a1 · · ·    .. .. .. . . . . . . satisfies

P

1, . . . , r 1, . . . , r

> 0,

r = 1, . . . , n,

which implies that P is also totally positive (see Theorem 4.6). It is known that if n ak xk p(x) = k=0

is a Hurwitz polynomial and q(x) =

m

bk xk

k=0

is a Hurwitz polynomial, then the polynomial r k=0

ak bk xk

4.11 Remarks

125

is also a Hurwitz polynomial, where r = min{n, m}. Thus the Hadamard product of two totally positive Hurwitz matrices of the above form is again a totally positive Hurwitz matrix of the appropriate form.

4.11 Remarks The main reference books on totally positive kernels and Chebyshev systems are Gantmacher, Krein [1950], Karlin [1968], Karlin, Studden [1966] and Krein, Nudel’man [1977]. The examples of Sections 4.2 through 4.6 can all be found, in more or less detail, in Gantmacher, Krein [1937]. The examples of Sections 4.2 and 4.3 can also be found in P´ olya, Szeg˝o [1976] (originally published in 1925). Karlin [1968] contains many, many additional examples of totally positive and strictly totally positive kernels and matrices. See also Carlson, Gustafson [1983] for some further examples. For a discussion of the theory of moments as presented in the proof of Theorem 4.4, see Karlin, Studden [1968], Chap. V, Krein, Nudel’man [1977], Chap. V and Shohat, Tamarkin [1943]. The form of the representation of the generating functions for totally positive infinite Toeplitz matrices was conjectured by Schoenberg in 1951. It was partially solved in Aissen, Schoenberg, Whitney [1952]. The final proof of the representations is to be found in Edrei [1952] and Edrei [1953]. The sequences in the totally positive Toeplitz matrices are also called Pólya Frequency Sequences; see Karlin [1968], Chap. 8, and references therein. For many more examples of sequences satisfying Theorem 4.5, see e.g., Brenti [1989], Brenti [1995], Pitman [1997], and Wang, Yeh [2005]. Theorem 4.6 is from Goodman, Sun [2004]. The fact that in the special case M = 2 (Hurwitz polynomials) the initial minors being strictly positive implies the total positivity of the full matrix was first proved by Asner [1970] and then reproved in a more transparent form by Kemperman [1982]. An example of a polynomial p whose zeros are not all in the closed left-hand plane Re (z) ≤ 0, but where the associated Hurwitz matrix P is totally positive (but without the strict positivity of the appropriate principal minors), can be found in Asner [1970]. The proof of Theorem 4.6 follows the lines of Kemperman’s proof. See also Holtz [2003] and references therein for another approach to the Hurwitz polynomials and the total positivity of the associated matrix. Material on Hurwitz matrices may be found in various texts, e.g., Gantmacher [1953], Marden [1966], and Rahman, Schmeisser [2002]. Theorem 4.8 is a special case of a more general result; see e.g., Gantmacher [1953], Theorem 13, Chap. XV of the English translation.

126

Examples

Theorem 4.10 and Proposition 4.11 can be found, for example, in Rahman, Schmeisser [2002], Cor. 10.6.13 (the notation is somewhat different). Applying the criteria of Theorem 2.16 we obtain relatively simple sufficient conditions for when certain of the matrices considered in this chapter are totally positive. For example, the Hankel matrix A = (bi+j )ni,j=0 is strictly totally positive if bk > 0, k = 0, 1, . . ., and π 2 b2 , k = 1, . . . , 2n − 1, bk−1 bk+1 > 4 cos n+2 k while the Toeplitz matrix A = (aj−i )∞ i,j=1 with ak = 0 for k < 0 and k > n is totally positive if ak > 0, k = 0, . . . , n, and a2k ≥ 4ak−1 ak+1 ,

k = 1, . . . , n − 1.

Thus this condition is sufficient to imply that the polynomial p(x) =

n

ak xk

k=0

has all real negative zeros. This was already proved in Kurtz [1992]. For a general discussion and survey of Hadamard products see Horn, Johnson [1991], Chap. 5, and the many references therein. It seems that Markham [1970] was the first to consider Hadamard products of totally positive matrices. He showed that the class of totally positive matrices is not closed under the operation of Hadamard products and proved Proposition 4.12. The Schur Product Theorem concerning the Hadamard product of positive definite matrices can be found, for example, in Horn, Johnson [1991], p. 309. Mal´ o’s Theorem is from Mal´ o [1895]. In Wagner [1992] the result concerning the Hadamard product of Toeplitz matrices is extended to a subclass of the totally positive Toeplitz matrices with an infinite number of nonzero coefficients. The closure of Hurwitz matrices under the Hadamard product is in Garloff, Wagner [1996a]. Its proof is far beyond the scope of this monograph. Some additional results can be found in Garloff, Wagner [1996b] and Crans, Fallat, Johnson [2001].

5 Eigenvalues and eigenvectors

In this chapter we review the spectral properties of totally positive matrices. A strictly totally positive matrix has positive, simple eigenvalues and the associated eigenvectors possess an intricate structure. Such is not the case for totally positive matrices. However, there is an intermediate set of matrices with the same spectral properties as strictly totally positive matrices. These matrices are called oscillation matrices. They shall be discussed in Section 5.1. In Section 5.2 we present the Gantmacher–Krein Theorem (Theorem 5.3) and give two quite different proofs thereof. This theorem contains the main spectral properties of oscillation matrices. In Section 5.3 we consider eigenvalues of the principal submatrices of such matrices and study their behaviour. We study in more detail the properties of eigenvectors of oscillation matrices in Section 5.4. Finally, in Section 5.5, we look at how the eigenvalues of oscillation matrices vary as functions of the elements of the matrix.

5.1 Oscillation matrices Oscillation matrices are a class of matrices intermediary between totally positive and strictly totally positive matrices. They share the eigenvalue and eigenvector structure of strictly totally positive matrices. Definition 5.1 An n × n matrix A is said to be an oscillation matrix if A is totally positive and some power of A is strictly totally positive. Importantly, there are relatively simple criteria for determining if a totally positive matrix is an oscillation matrix. 127

128

Eigenvalues and eigenvectors

Theorem 5.2 An n × n matrix A = (aij )ni,j=1 is an oscillation matrix if and only if A is totally positive, nonsingular, and ai,i+1 , ai+1,i > 0, i = 1, . . . , n − 1. Furthermore, if A is an oscillation matrix, then An−1 is strictly totally positive. Proof The necessity of the above conditions is rather simple. If A is an oscillation matrix then A is totally positive. If A is singular then every power of A is singular and thus no power of A can be strictly totally positive. Assume A is totally positive and nonsingular and ai,i+1 = 0 for some i. Then from Theorem 1.19 we have ars = 0 for all r ≤ i and s ≥ i+1. As is readily verified, this implies that the (r, s) elements of every power of A also vanish for all r ≤ i and s ≥ i + 1. This proves the necessity. The sufficiency is less elementary. We first prove that i1 , . . . , ir A >0 j1 , . . . , jr for all 1 ≤ i1 < · · · < ir ≤ n and 1 ≤ j1 < · · · < jr ≤ n satisfying |ik −jk | ≤ 1, k = 1, . . . , r, and max{ik , jk } < min{ik+1 , jk+1 }, k = 1, . . . , r − 1. In this proof we make use of the fact that since A is totally positive and nonsingular, every principal minor of A is strictly positive (see Theorem 1.13). Our proof of this claim is via induction on r. The result holds for r = 1 since all of the elements of A on the main and first off-diagonals are strictly positive. Assume the result holds for r − 1, but i1 , . . . , ir =0 A j1 , . . . , jr for some i1 , . . . , ir and j1 , . . . , jr , as above. Then we must have ik = jk for some k. From the induction hypothesis we have i2 , . . . , ir i1 , . . . , ir−1 ,A > 0. A j1 , . . . , jr−1 j2 , . . . , jr Thus, from Proposition 1.17, the totally positive matrix i1 , i1 + 1, . . . , ir B=A j1 , j1 + 1, . . . , jr is of rank r − 1. However, since ik = jk for some k, and from the conditions on the i1 , . . . , ir and j1 , . . . , jr , it follows that B contains an r × r singular principal submatrix of A. This is a contradiction since all principal minors of A are strictly positive. From the above it easily follows that some power of A is strictly totally

5.1 Oscillation matrices

129

positive. However we wish to prove a bit more; namely that An−1 is strictly totally positive. From Proposition 2.5 it suffices to prove that 1, . . . , r n−1 >0 A n − r + 1, . . . , n and

n−1

A

n − r + 1, . . . , n 1, . . . , r

>0

for r = 1, . . . , n. We will only consider the former inequalities. Note that i1 , . . . , ir n−1 A = j1 , . . . , jr 1 n−2 s1 , . . . , s1r s1 , . . . , srn−2 i1 , . . . , ir A 1 A 2 ···A s1 , . . . , s1r s1 , . . . , s2r j1 , . . . , jr where the sum is over all 1 ≤ s1 < · · · < sr ≤ n, = 1, . . . , n − 2. As each of these terms is nonnegative, for strict positivity to hold it suffices to prove that at least one of the products is strictly positive. For example, n−1 2 1 1 n−1 > 0, ···A A ≥A A n 3 2 n while

An−1

1, 2 n − 1, n

≥A

1, 2 1, 3

A

1, 3 2, 4

···A

n − 3, n − 1 n − 2, n

A

n − 2, n n − 1, n

> 0. The strict positivity is a consequence of what we proved above. Set s = (s1 , . . . , sr ), where sk = min{n − r + k, k + ( + k − r)+ },

k = 1, . . . , r,

= 0, . . . , n − 1.

Here m+ = max{m, 0}. It is now readily verified that |sk − s+1 k | ≤ 1 and +1 < sk+1 = min{sk+1 , s+1 max{sk , s+1 k } = sk k+1 }.

Thus

A

s1 , . . . , sr +1 s+1 1 , . . . , sr

> 0.

130


As s0 = (1, . . . , r) and sn−1 = (n − r + 1, . . . , n) it follows that 1, . . . , r n−1 > 0. A n − r + 1, . . . , n Remark If Ak , k = 1, . . . , n − 1, are arbitrary n × n oscillation matrices, then we also have that A1 · · · An−1 is strictly totally positive. The same proof applies.

5.2 The Gantmacher–Krein theorem The main theorem concerning spectral properties of oscillation matrices is the following. Theorem 5.3 (Gantmacher–Krein). Let A be an n × n oscillation matrix. Then the n eigenvalues of A are positive and simple. In addition, if we denote by uk a real eigenvector (unique up to multiplication by a nonzero constant) associated with the eigenvalue λk , where λ1 > λ2 > · · · > λn > 0, then     p p q − 1 ≤ S−  ci ui  ≤ S +  ci ui  ≤ p − 1, i=q

i=q

for each 1 ≤ q ≤ p ≤ n (and ci not all zero). In particular, S − (uk ) = S + (uk ) = k − 1 for k = 1, . . . , n. For the definitions of S − and S + , see Chapter 3. We present two very different proofs of Theorem 5.3. The first is the classic proof. In this classic proof we use two known general results. The first of these is Perron’s Theorem. Perron’s Theorem Let A be an n × n matrix, all of whose elements are strictly positive. Then A has a simple, positive eigenvalue that is strictly greater in modulus than all other eigenvalues of A. Furthermore the unique (up to multiplication by a nonzero constant) associated eigenvector may be chosen so that all its components are strictly positive. Proof There are numerous proofs of this result in the literature, and an almost uncountable number of generalizations. For completeness, we present a proof that seems to be one of the simpler and more transparent. For vectors x and y in Rn , we write x ≥ y if xi ≥ yi , i = 1, . . . , n, and

5.2 The Gantmacher–Krein theorem

131

x > y if xi > yi , i = 1, . . . , n. Set λ∗ = sup{λ : Ax ≥ λx , for some x ≥ 0} . Since all elements of A are strictly positive, we must have λ∗ > 0. A convergence (compactness) argument implies the existence of an x∗ ≥ 0 (x∗ = 0) such that Ax∗ ≥ λ∗ x∗ . If Ax∗ = λ∗ x∗ , then A(Ax∗ ) > λ∗ (Ax∗ ) since all entries of A are strictly positive. Setting y∗ = Ax∗ it follows that Ay∗ > λ∗ y∗ , which contradicts our definition of λ∗ . Thus Ax∗ = λ∗ x∗ . Now x∗ ≥ 0 (x∗ = 0) and thus Ax∗ > 0, whence x∗ > 0. We have found a positive eigenvalue with a strictly positive eigenvector. Let λ be any other eigenvalue of A. Then Ay = λy for some y ∈ Cn \{0}. Now |λ| |y| = |λy| = |Ay| ≤ A|y| , where |y| = (|y1 |, . . . , |yn |). From the definition of λ∗ , it follows that |λ| ≤ λ∗ . If |λ| = λ∗ , then we must have λ∗ |y| = A|y| (since otherwise λ∗ (A|y|) < A(A|y|), contradicting the definition of λ∗ ). Thus, for each i = 1, . . . , n, |λ| |yi | =

n j=1

aij yj =

n

aij |yj | .

j=1

This implies the existence of a γ ∈ C, |γ| = 1, such that γyj = |yj | for each j = 1, . . . , n. Thus, we may in fact assume that if |λ| = λ∗ , then λ = |λ| = λ∗ , and for the associated eigenvector y, we have y ≥ 0. Two consequences of these facts are the following: for every eigenvalue λ, λ = λ∗ , we have |λ| < λ∗ ; the geometric multiplicity of the eigenvalue λ∗ is exactly 1. The latter holds because, if not, we can easily construct a real eigenvector associated with λ∗ that is not of one sign, and this contradicts the above analysis. It remains to prove that the eigenvalue λ∗ is of algebraic multiplicity 1. Assume not. There then exists a vector y∗ (linearly independent of x∗ ) such that Ay∗ = λ∗ y∗ + αx∗ with some α = 0. This y∗ is called a “generalized eigenvector.” The transpose of A, namely AT , has the same eigenvalues as A and is obviously

132


strictly positive. As such there exists an eigenvector w∗ > 0 associated with the eigenvalue λ∗ . Now λ∗ (w∗ , y∗ ) = (AT w∗ , y∗ ) = (w∗ , Ay∗ ) = λ∗ (w∗ , y∗ ) + α(w∗ , x∗ ) . Since w∗ , x∗ > 0 and α = 0 we have α(w∗ , x∗ ) = 0, a contradiction. Thus the algebraic multiplicity of λ∗ is one. The second general result we use in the proof of Theorem 5.3 is called Kronecker’s Theorem. We recall from Chapter 1 that the p th compound matrix, denoted by A[p] , of the n × n matrix A is defined as the np × np matrix with entries (A(i, j))i∈Ipn ,j∈Ipn where the i ∈ Ipn and j ∈ Ipn are arranged in lexicographic order, i.e., for i, j ∈ Ipn we set i ≥ j (i = j) if the first nonzero term in the sequence i1 − j1 , . . . , ip − jp is positive. Kronecker’s Theorem Let A be an n × n matrix with eigenvalues λ1 , . . . , λn listed to their algebraic multiplicity. Then the np eigenvalues of A[p] , listed to their algebraic multiplicity, are λi1 · · · λip for 1 ≤ i1 < · · · < ip ≤ n. Proof We present two proofs. Every n × n matrix A may be written in the form A = P −1 T P where T is an upper triangular matrix. As such, the diagonal entries of T are simply the eigenvalues of A. Now, from the Cauchy–Binet formula, A[p] = (P −1 )[p] T[p] P[p] = (P[p] )−1 T[p] P[p] . The matrix T[p] is upper triangular and its diagonal entries are exactly the products of p distinct diagonal entries of T . This proves the result. A second proof is the following. Associated with each eigenvalue λi is an eigenvector/generalized eigenvector ui , i = 1, . . . , n, such that the u1 , . . . , un span Cn . Now (see Chapter 1) A[p] ui1 ∧ · · · ∧ uip = Aui1 ∧ · · · ∧ Auip from which it follows that the ui1 ∧ · · · ∧ uip are eigenvectors/generalized eigenvectors of A[p] with associated eigenvalues λi1 · · · λip . Since these vectors are linearly independent, we have determined all the np eigenvalues (and generalized eigenvectors) of A[p] .


133

We now present our first proof of Theorem 5.3. Proof of Theorem 5.3 It suffices to prove the theorem for strictly totally positive matrices. For if A has eigenvalues λ1 , . . . , λn , listed to their algebraic multiplicity, then Ak has the eigenvalues λk1 , . . . , λkn . If we show that λk1 > · · · > λkn > 0 for all k sufficiently large, then obviously we must have λ1 > · · · > λn > 0. In addition, if Au = λu then Ak u = λk u, so that if A has n distinct eigenvalues, then the eigenvectors of A and Ak are one and the same. Let λ1 , . . . , λn denote the eigenvalues of A listed to their algebraic multiplicity, and assume that |λ1 | ≥ · · · ≥ |λn | ≥ 0. As A is strictly totally positive we have that A[p] is a matrix all of whose entries are strictly positive. From the Perron and Kronecker Theorems applied to A[p] we have λ1 · · · λp > |λ1 · · · λp−1 λp+1 | for all p = 1, . . . , n. In addition, A is nonsingular and thus |λn | > 0. From these two facts we have λ1 > · · · > λn > 0 . From the Perron and Kronecker Theorems it also follows that by a suitable normalization of the associated eigenvectors u1 , . . . , un , we may assume that u1 ∧ · · · ∧ up > 0 for p = 1, 2, . . . , n. It remains for us to prove " the sign change properties of the eigenvectors p 1 n + i u , . . . , u . Assume S ≥ p for some choice of nontrivial i=q ci u (cq , . . . , cp ). There then exist j0 < · · · < jp and an ε ∈ {−1, 1} such that   p ε(−1)k  ci ui  ≥ 0 , k = 0, . . . , p . i=q

"p

jk

Set u0 = i=q ci ui . Let ui = (u1,i , . . . , un,i ), i = 0, 1, . . . , n, and U denote the matrix with columns u0 , u1 , . . . , un . Thus j0 , j1 , . . . , jp p = det (ujk ,i )i,k=0 = 0 U 0, 1, . . . , p since u0 is a linear combination of the u1 , . . . , up . On the other hand, when

134


expanding this matrix by the first column we obtain

p j , . . . , j , . . . , j j0 , j1 , . . . , jp 0 k p = . (−1)k ujk ,0 U U 0, 1, . . . , p 1, . . . , p k=0

1

Since u ∧ · · · ∧ u > 0, we have

j0 , . . . , jk , . . . , jp > 0, U 1, . . . , p p

for each k = 0, . . . , p. As (−1)k ujk ,0 is weakly of one sign, we must have ujk ,0 = 0, for all k = 0, . . . , p. But this is impossible since u1 ∧ · · · ∧ up > 0 implies that all p × p minors of the n × p matrix with columns u1 , . . . , up are nonsingular. In other words, no nontrivial linear combination of the u1 , . . . , un vanishes at p coordinates. Thus   p ci ui  ≤ p − 1. S+  i=q

Let v1 , . . . , vn be left eigenvectors of A with associated eigenvalues λ1 , . . . , λn , respectively. We assume, by what we have proved so far, that we have normalized the {vj }nj=1 so that v1 ∧ · · · ∧ vp > 0 for p = 1, . . . , n. Thus, in particular,   p bj v j  ≤ p − 1 S+  j=1

for every choice of nontrivial (b1 , . . . , bp ), p = 1, . . . , n. Let u ∈ Rn \{0} be any vector satisfying (u, vj ) = 0 ,

j = 1, . . . , q − 1 .

We claim that S − (u) ≥ q − 1. If S − (u) = r ≤ q − 2, then there exist indices 1 ≤ i1 < · · · < ir < n and an ε ∈ {−1, 1} such that ε(−1)k uj ≥ 0 ,

ik−1 + 1 ≤ j ≤ ik ,

k = 1, . . . , r + 1,

and uj = 0 for some ik−1 + 1 ≤ j ≤ ik , and each k = 1, . . . , r + 1. Here i0 = 0 and ir+1 = n. Let v=

r+1 j=1

bj v j


135

satisfy " v = 0 and vik = 0, k = 1, . . . , r. Such a v exists. Since r+1 + j S ≤ r, we must have j=1 bj v δ(−1)k vj > 0 ,

ik−1 + 1 ≤ j < ik ,

k = 1, . . . , r + 1,

and also δ(−1)r+1 vn > 0, where δ ∈ {−1, 1}. As r + 1 ≤ q − 1 and (u, v) = 0 , this is a contradiction, implying that S − (u) ≥ q − 1. The calculation λi (ui , vj ) = (λi ui , vj ) = (Aui , vj ) = (ui , vj A) = λj (ui , vj ), implies that (ui , vj ) = 0 for i = j. Thus   p  j = 1, . . . , q − 1 , ci ui , vj  = 0 , i=q

and therefore

  p ci ui  ≥ q − 1 , S−  i=q

which proves the theorem. Here is an alternative proof of this lower bound. The matrix A−1 is not a strictly totally positive matrix. However, C = DA−1 D is strictly totally positive where D is the diagonal matrix with diagonal entries alternately 1 and −1 (see Proposition 1.6). For u = (u1 , . . . , un ), set ! = Du = (u1 , −u2 , . . . , (−1)n+1 un ), i.e., component uj is replaced by u (−1)j−1 uj . From Lemma 3.1 S − (u) + S + (! u) = n − 1 for every u ∈ Rn , u = 0. As uk is an eigenvector of A with associated !k eigenvalue λk , the strictly totally positive matrix C has the eigenvector u with associated eigenvalue 1/λk . Because 1 1 > ··· > >0 λn λ1 it follows from our previous result that   p ! i  ≤ n − q, ci u S+  i=q

136


implying

  p ci ui  ≥ q − 1 . S−  i=q

We now present a very different proof of Theorem 5.3. The proof of the sign change properties of the eigenvectors will be based on the variation diminishing properties of strictly totally positive matrices. We start by providing an alternative proof of the fact that A has n simple and positive eigenvalues. We also simultaneously prove that the eigenvalues of the two principal submatrices obtained by deleting from A either the first row and column, or the last row and column, strictly interlace the eigenvalues of A. Proposition 5.4 Let A be an n × n strictly totally positive matrix. Then its n eigenvalues are positive and simple. In addition, if these eigenvalues (k) (k) are denoted by λ1 > λ2 > · · · > λn > 0, and µ1 > · · · > µn−1 > 0 are the eigenvalues of the principal submatrix of A obtained by deleting its kth row and column, then (k)

λj > µj

j = 1, . . . , n − 1 ,

> λj+1 ,

for k = 1 and k = n. Proof For ease of notation let Aλ = A − λI. From Sylvester’s Determinant Identity we have 2, . . . , n − 1 1, . . . , n Aλ Aλ 2, . . . , n − 1 1, . . . , n 1, . . . , n − 1 2, . . . , n = Aλ Aλ 1, . . . , n − 1 2, . . . , n 1, . . . , n − 1 2, . . . , n −Aλ Aλ . (5.1) 2, . . . , n 1, . . . , n − 1 It is readily verified that 1, . . . , n − 1 , Aλ 2, . . . , n

Aλ

2, . . . , n 1, . . . , n − 1

>0

(5.2)

for all λ > 0. Simply expand each determinant as a polynomial in λ and note that all the coefficients of this polynomial are strictly positive since all minors of A are strictly positive. We now use an induction argument to prove our result. The case n = 2


137

is easily checked by hand. As such we assume that n > 2. For notational ease, set 1, . . . , n p(λ) = Aλ 1, . . . , n 2, . . . , n q1 (λ) = Aλ , 2, . . . , n 1, . . . , n − 1 q2 (λ) = Aλ 1, . . . , n − 1 2, . . . , n − 1 . r(λ) = Aλ 2, . . . , n − 1 Thus from (5.1) and (5.2) p(λ)r(λ) < q1 (λ)q2 (λ)

(5.3)

for all λ > 0. We assume, by the induction hypothesis, that the n − 2 zeros of r are positive and simple, and interlace the n − 1 positive simple zeros of both q1 and q2 . Let µ1 > · · · > µn−1 > 0 (1)

denote the zeros of either q1 or q2 , i.e., µj = µj 1, . . . , n − 1. From (5.3) it follows that p(µi )r(µi ) < 0 ,

(n)

or µj = µj , j =

i = 1, . . . , n − 1 .

Furthermore, as r(0) > 0 and by the induction hypothesis, r(µi )(−1)i+n−1 > 0 ,

i = 1, . . . , n − 1 .

We therefore have p(µi )(−1)i+n > 0 ,

i = 1, . . . , n − 1 .

As p(0) > 0 and p(µn−1 ) < 0, the polynomial p has an additional zero in (0, µn−1 ). Furthermore, the polynomial p has leading coefficient (−1)n and p(µ1 )(−1)n+1 > 0. Thus p has an additional zero in (µ1 , ∞). As such, p has n−1 n positive, simple zeros that are interlaced by the {µj }j=1 . This advances the induction step. Another Proof of Theorem 5.3 We assume that λ > µ > 0 are eigenvalues of A with associated eigenvectors x and y. Then from Theorem 3.3 we have for all (α, β) = (0, 0) S + (λαx + µβy) = S + (A(αx + βy)) ≤ S − (αx + βy) .

(5.4)

138


Note that this implies, taking β = 0 and α = 0, respectively, that S + (x) = S − (x) and S + (y) = S − (y) . Assume that both α = 0 and β = 0. We may rewrite (5.4) as µ S + αx + βy ≤ S − (αx + βy) . λ Iterating this process we have µ k + αx + βy ≤ S − (αx + βy) , S λ for every positive integer k. As k → ∞, and since µ/λ < 1, we have from Lemma 3.2 µ k + lim S αx + βy ≥ S − (x). λ k→∞ (In fact, since S − (x) = S + (x) it follows that µ k + lim S αx + βy = S − (x) .) k→∞ λ Similarly, from (5.4) S + (αx + βy) ≤ S − and iterating this process we obtain +

S (αx + βy) ≤ S

−

µ λ

αx + βy ,

µ k λ

αx + βy

,

for every positive integer k. As k → ∞, and since µ/λ < 1 and S − (y) = S + (y), it follows (see Lemma 3.2) that µ k lim S − αx + βy ≤ S + (y) = S − (y) . k→∞ λ Thus for every (α, β) = (0, 0), S − (x) ≤ S − (αx + βy) ≤ S + (αx + βy) ≤ S − (y) . If S − (x) = S − (y), then equality holds throughout our sequence of inequalities, and in particular S + (αx + βy) = S − (αx + βy) for all (α, β) = (0, 0). This is impossible since there are choices of α, β for which the first or last coefficient of the vector αx + βy vanishes, in which case the above equality cannot possibly hold. Therefore S − (x) < S − (y).


139

We proved in Proposition 5.4 that A has n simple, positive eigenvalues and thus n associated eigenvectors. For each of these n eigenvectors we have S + (x) = S − (x) ∈ {0, 1, . . . , n − 1}, and if λ > µ > 0 are eigenvalues with associated eigenvectors x and y, then S − (x) < S − (y). This proves that if λ1 > · · · > λn > 0 , are the eigenvalues of A, and uk is a real eigenvector associated with the eigenvalue λk , then necessarily S + (uk ) = S − (uk ) = k − 1,

k = 1, . . . , n.

Now assume that 1 ≤ q ≤ p ≤ n and we are given real constants cq , . . . , cp , not all zero. Then, following the above analysis, repeated application of       p p p S+  ci λi ui  = S +  ci Aui  ≤ S −  ci ui  i=q

i=q

i=q

implies that for every positive integer m       m p p p λ i S+  ci ui  ≤ S −  ci ui  ≤ S +  ci ui  λ q i=q i=q i=q   m p λp ≤ S−  ci ui  . λ i i=q We may assume that cp , cq = 0, and apply Lemma 3.2 to obtain   m p λi + i ci u ≥ S − (uq ) = q − 1 , lim S λ m→∞ q i=q while



p

lim S − 

m→∞

Thus

ci

i=q

 q − 1 ≤ S− 

p i=q

which completes the proof.

λp λi

m



 ui  ≤ S + (up ) = p − 1 . 

ci ui  ≤ S + 

p i=q

 ci ui  ≤ p − 1 ,

140


Remark There is one more result that is sometimes stated in connection with the Gantmacher–Krein Theorem (Theorem 5.3). It has to do with the “interlacing of the zeros” of uk+1 and uk , which is important in the continuous (integral) analogue of this theorem. One form of this is the following. Let uk (t) be the continuous function defined on [1, n] which is linear on each [j, j + 1], j = 1, . . . , n − 1, and such that uk (j) is the jth component of uk . Since S − (uk ) = S + (uk ) = k − 1 , the function uk (t) has exactly k − 1 zeros that are each strict sign changes. Since k − 1 ≤ S − (αuk + βuk+1 ) ≤ S + (αuk + βuk+1 ) ≤ k , for every choice of (α, β) = (0, 0), it can be shown that the k − 1 zeros of uk (t) strictly interlace the k zeros of uk+1 (t). Since strictly totally positive matrices are dense in the set of totally positive matrices, see Theorem 2.6, and because of the continuity of the eigenvalues as functions of the matrix entries, we have the following: Corollary 5.5 The eigenvalues of totally positive matrices are both real and nonnegative.

5.3 Eigenvalues of principal submatrices What can we say about the eigenvalues of the principal submatrices of a strictly totally positive matrix? Let λ1 > · · · > λn > 0 denote the eigenvalues of A, and (k)

(k)

µ1 > · · · > µn−1 > 0 the eigenvalues of the principal submatrix of A obtained by deleting its kth row and column. As we have seen in Proposition 5.4, the n − 1 positive, (1) n−1 (n) n−1 simple eigenvalues {µi }i=1 and {µi }i=1 strictly interlace those of the original matrix A. This naturally begs the question of whether this same property holds for all principal submatrices? It also easily follows (see the proof of the Perron Theorem on positive matrices) that (k)

λ1 > µ1

5.3 Eigenvalues of principal submatrices

141

for each k. However, the strict interlacing property does not hold for all principal submatrices. As an example, consider   1 1 0 A =  2 2 1 . 2 2 1 The matrix A is totally positive but not strictly totally positive. However, since strictly totally positive matrices are dense in the class of totally positive matrices, there are strictly totally positive matrices whose eigenvalues (and the eigenvalues of its principal submatrices) √are arbitrarily √ close to those of A. The eigenvalues of A are 2 + 3, 2 − 3, and 0. The eigenvalues of the principal submatrix of A obtained by deleting the second row and column are 1 and 1. Obviously we do not have interlacing. Nevertheless there is a weaker interlacing that does hold and it is the following. Theorem 5.6 Let A be an n × n strictly totally positive matrix. Then for each k, 1 < k < n, (k)

λj−1 > µj

j = 1, . . . , n − 1 ,

> λj+1 ,

(where λ0 = λ1 ). Proof We first prove that for any k and j as above, (k)

µj

> λj+1 .

Let Ak denote the principal submatrix of A obtained by deleting the kth (k) row and column. Let v denote a real eigenvector of Ak with eigenvalue µj , i.e., (k)

Ak v = µj v . From Theorem 5.3 S + (v) = S − (v) = j − 1 . We construct the vector v ∈ Rn from the vector v ∈ Rn−1 by simply inserting a 0 as its new kth coordinate. That is, we write (somewhat abusing notation) v = (v1 , . . . , vk−1 , vk+1 , . . . , vn ) and set v = (v1 , . . . , vk−1 , 0, vk+1 , . . . , vn ) .

142


Let v be defined by (k)

Av = µj v . Thus v = (v1 , . . . , vk−1 , wk , vk+1 , . . . , vn ) , for some easily calculated wk . From Theorem 3.3 and the properties of v, v and v we have, j − 1 = S + (v) ≤ S + (v ) ≤ S − (v ) = S − (v) = j − 1 . Thus S + (v ) = S − (v ) = j − 1 . Let u denote a real eigenvector of A with eigenvalue λj+1 , i.e., Au = λj+1 u . Then from Theorem 5.3 S + (u) = S − (u) = j . If uk = 0, then the vector obtained by deleting uk from u is an eigenvector of Ak , also with eigenvalue λj+1 . Since this new vector also has exactly j sign changes, it follows, again from Theorem 5.3, that (k)

(k)

λj+1 = µj+1 < µj . On the other hand, if wk = 0 then v = v is an eigenvector of A with j − 1 sign changes. As such (k)

µj

= λj > λj+1 .

Thus we may now assume that both uk and wk are nonzero. The vectors u and v are defined up to multiplication by a nonzero constant. As such we assume that uk and wk are both positive and set c∗ = inf{c : c > 0, S − (cv + u) ≤ j − 1} . We claim that c∗ is a well-defined positive number. This is a consequence of the following. For all c sufficiently large, e.g., such that c|vi | > |ui | if vi = 0, we have S − (cv + u) ≤ S + (cv + u) ≤ S + (v ) = j − 1 . (Here we have used the fact that cvk + uk = uk and wk are of the same

5.3 Eigenvalues of principal submatrices

143

sign.) For all c sufficiently small and positive, e.g., such that c|vi | < |ui | if ui = 0, we have S − (cv + u) ≥ S − (u) = j . From the definition of c∗ and continuity properties of S + and S − , it follows that S − (c∗ v + u) ≤ j − 1 and S + (cv + u) ≥ j for all c ≤ c∗ . Now (k)

A(c∗ v + u) = c∗ µj v + λj+1 u . Let (k)

c=

c∗ µj > 0. λj+1

From Theorem 3.3, S + ( cv + u) ≤ S − (c∗ v + u) ≤ j − 1 . Since cv + u and cv + u differ only in their kth coordinates, where both are positive, it follows that S + ( cv + u) = S + ( cv + u) ≤ j − 1 . (k)

This implies, from the above, that c > c∗ . Thus µj The proof of the reverse inequality

> λj+1 .

(k)

λj−1 > µj

for any k, and j = 2, . . . , n − 1, is essentially the same. Let x be a real eigenvector of A with eigenvalue λj−1 . That is, Ax = λj−1 x . Note that S + (x) = S − (x) = j − 2 . (k)

The vectors v, v , and v are as defined above. If wk = 0 then µj (k)

(k)

= λj
µj . We may thus assume that wk and xk are both positive. We set a∗ = inf{a : a > 0, S − (ax + v ) ≤ j − 2} ,

144


and now follow the previous reasoning, essentially verbatim. As strictly totally positive matrices are dense in the set of totally positive matrices and because of the continuity of the eigenvalues as functions of the matrix entries, we have the following. Corollary 5.7 Let A be an n × n totally positive matrix with eigenvalues λ1 ≥ · · · ≥ λn ≥ 0 (listed to their algebraic multiplicity). Let (k)

(k)

µ1 ≥ · · · ≥ µn−1 ≥ 0 denote the eigenvalues of the principal submatrix of A obtained by deleting its kth row and column. Then for any k, 1 < k < n, (k)

λj−1 ≥ µj

≥ λj+1 ,

j = 1, . . . , n − 1 ,

(where λ0 = λ1 ).

5.4 Eigenvectors In this section we study in considerably more detail the eigenvector structure of a strictly totally positive or oscillation matrix. We have from Theorem 5.3 that if uk is a real eigenvector associated with the kth eigenvalue in magnitude of an oscillation matrix, then     p p ci ui  ≤ S +  ci ui  ≤ p − 1 q − 1 ≤ S−  i=q

i=q

for 1 ≤ q ≤ p ≤ n and nontrivial (cq , . . . , cp ). Given vectors u1 , . . . , un satisfying the above inequalities, are they necessarily the eigenvectors of some oscillation and thus strictly totally positive matrix? In other words, do the above inequalities exactly characterize the set of eigenvectors of strictly totally positive matrices? In addition, what do these inequalities imply about the vectors uk ? That is, what can we say about the eigenmatrix U whose columns are the uk ? We prove the following. Theorem 5.8 Assume we are given vectors u1 , . . . , un in Rn . Let U be the n × n matrix whose kth column is uk . The following are equivalent:

5.4 Eigenvectors

145

(i) There exists an n × n strictly totally positive matrix A with eigenvalues λ1 > · · · > λn > 0 and associated eigenvectors u1 , . . . , un , respectively. (ii) For every 1 ≤ q ≤ p ≤ n and nontrivial (cq , . . . , cp )     p p q − 1 ≤ S−  (5.5) ck uk  ≤ S +  ck uk  ≤ p − 1. k=q

k=q

(iii) There exist εp ∈ {−1, 1}, p = 1, . . . , n, such that i1 , . . . , ip εp U >0 1, . . . , p

(5.6)

for all 1 ≤ i1 < · · · < ip ≤ n and p = 1, . . . , n, and δq ∈ {−1, 1}, q = 1, . . . , n, such that n iq , . . . , in (−1) k=q ik > 0 (5.7) δq U q, . . . , n for all 1 ≤ iq < · · · < in ≤ n and q = 1, . . . , n. If we assume that εn = (−1)n(n+1)/2 (δ1 = 1), then εp δp+1 = (−1)p(p+1)/2 , p = 1, . . . , n. Proof (i) ⇒ (ii) is contained in Theorem 5.3. (ii) ⇒ (iii). We first prove (5.6). The result holds for p = 1 by inspection since S + (u1 ) = S − (u1 ) = 0. We assume, by induction, that the result holds for p − 1 and prove it for p, where p ≥ 2. If i1 , . . . , ip =0 U 1, . . . , p for some 1 ≤ i1 < · · · < ip ≤ n then there exists a nontrivial (c1 , . . . , cp ) such that p

k ck u = 0, j = 1, . . . , p. k=1

ij

But then

S

+

p

k

ck u

≥p

k=1

which is a contradiction. Thus i1 , . . . , ip = 0. U 1, . . . , p

146


It remains to prove that all such minors are of one sign. Choose 1 ≤ i1 < · · · < ip−1 ≤ n. Let (c1 , . . . , cp ) be a nontrivial vector of coefficients satisfying p

k ck u = 0, j = 1, . . . , p − 1. k=1

ij

This implies that

S+

p

≥ p − 1,

ck uk

k=1

and from (5.5) we must have S

+

p

= p − 1.

k

ck u

k=1

Thus cp = 0, and for ij−1 < r < ij , where i0 = 0, ip = n + 1, p

j k ck u > 0, j = 1, . . . , p, ε(−1) k=1

r

for some ε ∈ {−1, 1}. Solve for cp using the p equations

p ck uk = 0, j = 1, . . . , p − 1, k=1

and

ij

p

k=1

k

ck u

= αr . r

For any r ∈ {1, . . . , n}\{i1 , . . . , ip−1 }, assuming ij−1 < r < ij , we have i1 , . . . , ip−1 αr U 1, . . . , p − 1 . cp = i1 , . . . , ij−1 , r, ij , . . . , ip−1 (−1)j+p U 1, . . . , p As sgn αr = (−1)j ε for some ε ∈ {−1, 1} it follows, using the induction hypothesis, that for some εp ∈ {−1, 1} i1 , . . . , ij−1 , r, ij , . . . , ip−1 >0 εp U 1, . . . , p for all r between ij−1 and ij and all j = 1, . . . , p, i.e., for all r ∈ {1, . . . , n}\{i1 , . . . , ip−1 }

5.4 Eigenvectors

147

with the rows arranged in increasing order. From a connectedness argument this implies that i1 , . . . , ip >0 εp U 1, . . . , p for all 1 ≤ i1 < · · · < ip ≤ n. We now prove (5.7). For u = (u1 , u2 , . . . , un ) we set ! = (u1 , −u2 , . . . , (−1)n+1 un ) u and recall that S − (u) + S + (! u) = n − 1 for every u ∈ Rn , u = 0, see Lemma 3.1. Thus the inequality   n ck uk  ≥ q − 1, S−  k=q

is equivalent to

 S+ 

n

 ! k  ≤ n − q, ck u

k=q

for nontrivial (cq , . . . , cn ). This is exactly the same inequality as assumed in the previous paragraph. As such there exists a δ!q ∈ {−1, 1} such that iq , . . . , in ! ! >0 δq U q, . . . , n ! denotes the n × n matrix with for all 1 ≤ iq < · · · < in ≤ n, where U k ! . This immediately translates into columns u n iq , . . . , in (−1) k=q ik +1 > 0, δ!q U q, . . . , n implying

δq U

iq , . . . , in q, . . . , n

n

(−1)

k=q ik

>0

for some δq ∈ {−1, 1} and all 1 ≤ iq < · · · < in ≤ n. Multiplying any one of the uk by a nonzero constant we may and will assume that 1, . . . , n = (−1)n(n+1)/2 , U 1, . . . , n

148


i.e., εn = (−1)n(n+1)/2 and δ1 = 1. Now for any p ∈ {1, . . . , n − 1}, by the Laplace expansion by minors, (−1)n(n+1)/2 = det U n ip+1 , . . . , in i1 , . . . , ip ik +k k=p+1 U = (−1) U 1, . . . , p p + 1, . . . , n 1≤i1 0

it follows that εp δp+1 = (−1)p(p+1)/2 . (iii) ⇒ (i). Set V = U −1 where, as above, we assume, without loss of generality, that det U = (−1)n(n+1)/2 . Then p 1, . . . , p j1 , . . . , jn−p (−1)n(n+1)/2 V = (−1) k=1 k+jk U p + 1, . . . , n j1 , . . . , jp where the j1 < · · · < jp and j1 < · · · < jn−p are complementary indices in {1, . . . , n}. As n−p j1 , . . . , jn−p (−1) k=1 jk > 0 δp+1 U p + 1, . . . , n

we have

p(p+1)/2

δp+1 (−1)

V

1, . . . , p j1 , . . . , jp

>0

for all 1 ≤ j1 < · · · < jp ≤ n and all p = 1, . . . , n − 1, i.e., 1, . . . , p εp V >0 j1 , . . . , jp for all 1 ≤ j1 < · · · < jp ≤ n and all p = 1, . . . , n. This implies that i1 , . . . , ip 1, . . . , p U V >0 j1 , . . . , jp 1, . . . , p

(5.8)

for every choice of 1 ≤ i1 < · · · < ip ≤ n and 1 ≤ j1 < · · · < jp ≤ n, and all p.

5.5 Eigenvalues as functions of matrix elements

149

For any λ1 > · · · > λn > 0 let Λ be the n × n diagonal matrix with diagonal entries {λ1 , . . . , λn }. Consider A = U ΛV. The columns of U are the (right) eigenvectors of A with associated eigenvalues λ1 , . . . , λn . We will choose the λj dependent upon U and V . To prove that A is strictly totally positive it suffices to show that i1 , . . . , ip >0 A j1 , . . . , jp for all choices of 1 ≤ i1 < · · · < ip ≤ n and 1 ≤ j1 < · · · < jp ≤ n. For A as above we have from the Cauchy–Binet formula i1 , . . . , ip k1 , . . . , kp i1 , . . . , ip = U λk 1 · · · λ k p V . A j1 , . . . , jp k1 , . . . , kp j1 , . . . , jp 1≤k1 λn > 0 so that A is strictly totally positive. (From the above construction we see that for any choice of λ1 > · · · > λn > 0 and A = U ΛV , we have that Am = U Λm V is a strictly totally positive matrix for m sufficiently large.)

5.5 Eigenvalues as functions of matrix elements How do the eigenvalues of an oscillation matrix vary as functions of the elements of the matrix? This is the topic of this section. We start with a general result. Proposition 5.9 Let A = (aij ) be an n × n matrix with distinct simple eigenvalues λ1 , . . . , λn and associated eigenvectors u1 , . . . , un . Let U = (uij ) be the n × n eigenmatrix whose kth column is uk . Set V = (vij ) where V = U −1 , and let vk denote the kth row of V . Then ∂λk = vki ujk ∂aij for all i, j, k ∈ {1, . . . , n}. Proof The {λk } are locally differentiable functions of the aij since they are the distinct roots of a polynomial of degree n whose coefficients depend algebraically upon the aij .

150


From the above Auk = λk uk , and thus n

ars usk = λk urk ,

r = 1, . . . , n.

s=1

Similarly vk A = λk vk , i.e., n

vkr ars = λk vks ,

s = 1, . . . , n.

r=1

Now A = U ΛV where Λ is the n × n diagonal matrix with diagonal entries {λ1 , . . . , λn }. Thus V AU = Λ, i.e., λk =

n

vkr ars usk ,

k = 1, . . . , n.

r,s=1

Differentiating the above with respect to aij , we have ∂λk ∂aij

=

=

n ∂vkr ∂usk ars usk + vkr ars ∂a ∂aij ij r,s=1

n

n n n ∂usk ∂vkr vki ujk + ars usk + vkr ars . ∂a ∂aij ij r=1 s=1 s=1 r=1

vki ujk +

Substituting from above gives ∂λk ∂aij

since

"n r=1

n ∂vkr

n

∂usk λk vks ∂a ∂aij ij r=1 s=1 n

n ∂vkr ∂usk = vki ujk + λk urk + vks ∂aij ∂aij r=1 s=1 n

∂ vkr urk = vki ujk + λk ∂aij r=1 = vki ujk =

vki ujk +

λk urk +

vkr urk ≡ 1 as V = U −1 .

An application of Proposition 5.9 leads to the following.

5.5 Eigenvalues as functions of matrix elements

151

Theorem 5.10 Let A = (aij ) be an n × n oscillation matrix with eigenvalues λ1 > · · · > λn > 0. Then ∂λn ∂λ1 > 0, (−1)i+j > 0, ∂aij ∂aij

i, j = 1, . . . , n,

and ∂λk ∂λk ∂λk ∂λk > 0, k = 1, . . . , n. > 0, > 0, (−1)k+1 > 0, (−1)k+1 ∂a11 ∂ann ∂a1n ∂an1 Proof Let U be as in Proposition 5.9. Without loss of generality we assume that u1k > 0, k = 1, . . . , n. This implies, from Theorem 5.3, that uk1 > 0, k = 1, . . . , n, and (−1)k+1 ukn > 0, (−1)k+1 unk > 0, k = 1, . . . , n. Set V = U −1 . Then the exact same inequalities hold for V , i.e., v1k , vk1 > 0, k = 1, . . . , n, and (−1)k+1 vkn , (−1)k+1 vnk > 0, k = 1, . . . , n. Theorem 5.10 is now an immediate consequence of these inequalities and the previous Proposition 5.9. The inequality ∂λ1 >0 ∂aij is also a consequence of Perron’s Theorem. Another consequence of Perron’s Theorem is the following. Let   a11 · · · a1n ..   ..  . .     Ac =  cak1 · · · cakn     . . . .  . .  an1

···

ann

i.e., Ac is obtained from A by multiplying the elements of the kth row by c. Proposition 5.11 Let A be an n × n oscillation matrix. For c > 0, let λ1 (c) > · · · > λn (c) > 0 denote the eigenvalues of Ac . Then ∂λ1 (c) > 0, ∂c

∂λn (c) >0 ∂c

and ∂(λ1 (c) · · · λr (c)) > 0, ∂c

r = 1, . . . , n.

152


Proof All three inequalities are consequences of Perron’s Theorem. That is, we use the characterization λ1 = sup{λ : Ax ≥ λx, for some x ≥ 0} valid for any strictly positive matrix. The first inequality is a direct application thereof (and is also in Theorem 5.10). The second inequality follows from an application of Perron’s Theorem to the oscillation matrix DA−1 c D. For a proof of the third inequality, apply Perron’s Theorem to the rth compound matrix A[r] . It is not necessarily true that ∂λr (c) >0 ∂c for all r. The 3 × 3 totally positive matrix   1 1 0 Ac =  2c 2c c  2 2 1 is such that λ2 (c) is a strictly decreasing function of c, c > 0.

5.6 Remarks The study of the spectral properties of integral equations with totally positive continuous kernels substantially predates the study of the spectral properties of totally positive matrices. In 1918 O. D. Kellogg proved the main spectral properties in the case of a symmetric continuous totally positive kernel (see Kellogg [1918]). Kellogg was an American mathematician who obtained his doctorate from G¨ ottingen in 1903 under the supervision of Hilbert. He is best known for his work on potential theory, and his book thereon Kellogg [1929] has been reprinted many times since. Both Krein and Gantmacher were very much influenced by Kellogg’s work on this subject. An announcement of the parallel result for continuous non-symmetric totally positive kernels is in Gantmacher [1936]. The main results concerning spectral properties of totally positive matrices are in Gantmacher, Krein [1937]. An announcement appears in Gantmacher, Krein [1935]. Oscillation matrices were introduced in Gantmacher, Krein [1937], and Theorem 5.2 can be found therein on p. 454 (see also Gantmacher, Krein [1950], Chap. II, §7, Karlin [1968], Chap. 2, §9, and Ando [1987]). The proof of Theorem 5.3, based on the Perron and Kronecker theorems, is from Gantmacher, Krein [1937], Theorems 10

5.6 Remarks

153

and 14. The same proof also appears in Gantmacher [1953], Gantmacher, Krein [1950], and Ando [1987]. The strict interlacing of the eigenvalues of a strictly totally positive matrix with the eigenvalues of the principal submatrices obtained by deleting the first (or last) row and column was first proved in Gantmacher, Krein [1937]. The proof as given in Proposition 5.4 is from Koteljanskii [1955]. The second proof of Theorem 5.3, based on variation diminishing, is from Elias, Pinkus [2002], which contains generalizations of Theorem 5.3 to the nonlinear setting (see also Pinkus [1985a], [1985b] and Buslaev [1990]). For more on the eigenvalues and eigenvectors of oscillation matrices see Eveson [1996], Karlin [1965], Karlin [1972] and Karlin, Pinkus [1974]. Fallat, Gekhtman, Johnson [2000] consider the possible spectral structure of a subclass of totally positive matrices. Theorem 5.6 is due to Pinkus [1998]. The example of a totally positive matrix where interlacing of the eigenvalues of the minor fails is from Karlin, (k) Pinkus [1974]. Friedland [1985] had previously proved that µ1 ≥ λ2 and (k) µn−1 ≥ λn for k = 1, . . . , n − 1. Parts of Theorem 5.8, in a slightly different form, appear in Gantmacher, Krein [1937], Theorem 16. Theorem 5.10 is also from Gantmacher, Krein [1937], Theorems 18 and 19. A survey of the spectral properties of totally positive kernels and matrices, with extensive references, can be found in Pinkus [1996].

6 Factorizations of totally positive matrices

In Chapter 2 (Theorem 2.12) we proved that every n × n totally positive matrix can be factored in the form A = LU where L and U are totally positive matrices, L being a lower triangular matrix and U an upper triangular matrix. In this chapter we study factorizations of totally positive matrices in considerably greater detail. We prove that if A is an n × n totally positive matrix, then A can always be factored in the form A = L1 · · · Ln−1 U 1 · · · U n−1 where each Lk is a 1-banded, lower triangular, totally positive matrix and each U k is a 1-banded, upper triangular, totally positive matrix. In fact there are generally an uncountable number of such factorizations, and at least 22n−2 different factorizations with a maximum number of zero entries. In Section 6.2, after some preliminaries, we start with one such factorization for A strictly totally positive by essentially writing down the factorization, and then proving its validity. We then show, using elementary operations, how it is possible to construct many other such factorizations. In Section 6.3 we generalize these results to totally positive matrices.

6.1 Preliminaries We start with two definitions to set our notation. Definition 6.1 For each i, j ∈ {1, . . . , n}, i = j, we define Ei,j (α) 154

6.1 Preliminaries

155

as the n × n unit diagonal matrix with α in the (i, j) position and zeroes in the other off-diagonal entries. We recall that right multiplication of A by Ei,j (α) is the operation whereby α times the ith column of A is added to the jth column of A with the other columns left unchanged. Left multiplication of A by Ei,j (α) is the operation whereby α times the jth row of A is added to the ith row of A with the other rows left unchanged. In addition, as is readily verified, (Eij (α))−1 = Eij (−α). Definition 6.2 We say that a lower triangular matrix L = (ij ) is r-banded if ij = 0 for i − j > r. Thus L has the form

L



      =      

1 × × × ×

 · · · · · · · · · ·

· · · · × /

· · · ×

· · × 01

· × 1 2

      .     

r

If L is an n × n matrix and, for whatever reason, we say that L is r-banded and lower triangular, where r ≥ n − 1, then this statement is meaningless and should be so understood. We list three elementary properties of r-banded matrices for easy reference. Proposition 6.3 If L is an r-banded lower triangular matrix and M is an s-banded lower triangular matrix, then LM is an (r + s)-banded lower triangular matrix. Proposition 6.4 A 1-banded lower triangular matrix is totally positive if and only if all its elements are nonnegative. We repeatedly use the following (contrast it with Proposition 6.3). Proposition 6.5 For k = 1, . . . , n − 1, let C k = E2,1 (α1 ) · · · Ek+1,k (αk ).

156

Factorizations of totally positive matrices

Then C k is a 1-banded unit diagonal lower triangular matrix with k C i+1,i = αi , i = 1, . . . , k, k C i+1,i = 0, i = k + 1, . . . , n − 1. If Rk = En−k+1,n−k (αn−k ) · · · En,n−1 (αn−1 ), then Rk is a 1-banded unit diagonal lower triangular matrix with k i = 1, . . . , n − k − 1, R i+1,i = 0, k R i+1,i = αi , i = n − k, . . . , n − 1. Note that when reversing the multiplication order (in C k ) we get Ek+1,k (α1 ) · · · E2,1 (αk ), which is a k-banded matrix, whose (i, j)th entry, 1 ≤ j < i ≤ k, is exactly αj · · · αi−1 , with other off-diagonal entries being zero.

6.2 Factorizations of strictly totally positive matrices We recall (Theorem 2.10) that an n × n strictly totally positive matrix A has a unique factorization of the form A = LDU where L is a unit diagonal lower strictly totally positive matrix, U is a unit diagonal upper strictly totally positive matrix, and D is a diagonal matrix whose diagonal entries are strictly positive. We also recall that the unit diagonal matrix L is lower strictly totally positive if it is lower triangular and i1 , . . . , ik >0 L j1 , . . . , jk whenever im ≥ jm , m = 1, . . . , k. We first prove a fundamental result from which all other factorization results will follow. Theorem 6.6 Let L be an n × n unit diagonal lower triangular matrix.

6.2 Factorizations of strictly totally positive matrices

157

Then L is lower strictly totally positive if and only if L can be factored in the form L = R1 · · · Rn−1

(6.1)

where Rk = En−k+1,n−k (αk,n−k ) · · · En,n−1 (αk,n−1 ),

k = 1, . . . , n − 1, (6.2)

with αk,j > 0, j = n − k, . . . , n − 1, k = 1, . . . , n − 1. Proof Assume L has the form (6.1) where each Rk satisfies (6.2), with the appropriate αk,j strictly positive. As L is the product of unit diagonal lower totally positive matrices, it is necessarily a unit diagonal lower totally positive matrix. But we want to prove that it is a lower strictly totally positive matrix. We recall (Theorem 2.8) that to prove that L is lower strictly totally positive it suffices to prove that i + 1, . . . , i + r L > 0, i = 0, 1, . . . , n − r, r = 1, . . . , n. 1, . . . , r (Since L is unit diagonal, the equations in the case i = 0 are of no interest.) In fact (see Proposition 2.9), we need only prove the strict positivity of the above inequalities for i = n − r. However, in the proof of the converse direction we make use of all these inequalities, and so we also prove them here. Recall that   1  0 ·      · ·     · ·     Rk =  0 · .     αk,n−k ·     · ·     · · αk,n−1

1

Because of the format (zero elements) of the Rk it easily follows that (L)i+1,1 = (Rn−i )i+1,i · · · (Rn−1 )2,1 ,

i = 1, . . . , n − 1.

As (Rn−i )i+1,i > 0,

i = 1, . . . , n − 1,

158


we have i = 1, . . . , n − 1.

(L)i+1,1 > 0, Now consider

L

i + 1, i + 2 1, 2

.

Again, from the form of the R1 , . . . , Rn−1 , it follows that i + 1, i + 2 i + 1, i + 2 2, 3 · · · Rn−1 , i = 1, . . . , n − 2. = Rn−i L i, i + 1 1, 2 1, 2 By assumption, i + 1, i + 2 n−i = (Rn−i )i+1,i (Rn−i )i+2,i+1 > 0, R i, i + 1 Thus

L

i + 1, i + 2 1, 2

i = 1, . . . , n − 2.

> 0,

i = 1, . . . , n − 2.

We progress in this fashion to obtain i + 1, . . . , i + r L > 0, i = 1, . . . , n − r, r = 1, . . . , n − 1. 1, . . . , r Thus L is lower strictly totally positive. Let us now assume that L is lower strictly totally positive. For k = 1, . . . , n − 1 we define the unit diagonal, 1-banded matrix Rk with k R i+1,i = 0, i = 1, . . . , n − k − 1, k R i+1,i > 0, i = n − k, . . . , n − 1 in the following manner. We will determine the (Rk )i+1,i , i = n − k, . . . , n − 1, k = 1, . . . , n − 1, from the previous sets of equations. That is, the equations (L)i+1,1 = (Rn−i )i+1,i · · · (Rn−1 )2,1 ,

i = 1, . . . , n − 1,

and the fact that (L)i+1,i > 0 define for us the strictly positive values (Rn−i )i+1,i ,

i = 1, . . . , n − 1.

The equations i + 1, i + 2 2, 3 i + 1, i + 2 · · · Rn−1 , i = 1, . . . , n − 2, = Rn−i L i, i + 1 1, 2 1, 2


159

and the strict positivity of the left-hand side of the equations define for us the strictly positive values i + 1, i + 2 = (Rn−i )i+1,i (Rn−i )i+2,i+1 , i = 1, . . . , n − 2, Rn−i i, i + 1 (since we set (Rn−i )i+2,i = 0), which in turn define for us the strictly positive values (Rn−i )i+2,i+1 ,

i = 1, . . . , n − 2.

We continue in this fashion to define all the desired (Rk )i+1,i . We claim that L = R1 · · · Rn−1 . That is, we have chosen the values of the Rk so that the formulæ for the i + 1, . . . , i + r L , i = 1, . . . , n − r, r = 1, . . . , n − 1, 1, . . . , r are valid. But do these formulæ necessarily guarantee that L = R1 · · · Rn−1 ? The answer is, of course, yes. The above set of data, and their strict positivity, determine explicitly all the entries of L. That is, the first column of L is given (r = 1). Given the first column (and the fact that its entries are not zero) and i + 1, i + 2 L , i = 1, . . . , n − 2, 1, 2 determines the entries of the second column of L, etc. . . Thus we do have L = R1 · · · Rn−1 . This proves the theorem. Remark From the above follow very explicit formulæ for the (Rk )i+1,i , i = n − k, . . . , n − 1, k = 1, . . . , n − 1, in terms of minors of L. But we shall not list them here. Let us now consider how we can construct a different factorization for L. Corollary 6.7 Let L be an n × n unit diagonal lower triangular matrix. Then L is lower strictly totally positive if and only if L can be factored in the form L = C n−1 · · · C 1

(6.3)

160


where C k = E2,1 (βk,1 ) · · · Ek+1,k (βk,k )

(6.4)

with βk,j > 0, j = 1, . . . , k, k = 1, . . . , n − 1. Proof There are two simple ways of justifying this corollary. The first is to take the proof of Theorem 6.6 and make the simple obvious modifications therein. The second explanation is more satisfying. Recall from Propositions 1.2 and 1.3 that a matrix A is totally positive (strictly totally positive) if and only if AT is totally positive (strictly totally positive) if and only if QAQ is totally positive (strictly totally positive), where   0 ··· 0 1  0 ··· 1 0    Q= . . . . . ... ...   .. 1

···

0 0

Note that QAQ is the matrix A with the order of its rows and columns reversed. Given a lower triangular matrix L = (ij )ni,j=1 , then (QLQ)T = QLT Q = (n−j+1,n−i+1 )ni,j=1 is also lower triangular. In addition, L is lower strictly totally positive if and only if (QLQ)T is lower strictly totally positive. Applying Theorem 6.6 to (QLQ)T gives us (QLQ)T = R1 · · · Rn−1 for R1 , . . . , Rn−1 as in the statement of Theorem 6.6. Thus L = (QRn−1 Q)T · · · (QR1 Q)T . Set C k = (QRk Q)T ,

k = 1, . . . , n − 1.

The C k are as in the statement of Corollary 6.7. We can and should look at the factorizations of Theorem 6.6 and Corollary 6.7 from a somewhat different (and historically more valid) perspective. To explain, let S k = Rk · · · Rn−1 ,

k = 1, . . . , n − 1.


161

Thus L = R1 · · · Rk−1 S k . As may be seen, S k is a full (n − k)-banded matrix (more on the structure of the S k anon) and S k = Rk S k+1 . To explain what is happening recall that (Ei,j (α))−1 = Ei,j (−α) and Rk = En−k+1,n−k (αk,n−k ) · · · En,n−1 (αk,n−1 ). Thus (Rk )−1 = En,n−1 (−αk,n−1 ) · · · En−k+1,n−k (−αk,n−k ), and S k+1 = En,n−1 (−αk,n−1 ) · · · En−k+1,n−k (−αk,n−k )S k . As previously noted, left multiplication of a matrix A by Er+1,r (−α) is the operation whereby α times the rth row of A is subtracted from the r + 1st row of A, with the other rows left unchanged. Thus S k+1 is obtained from S k by successively eliminating (making zero) the k strictly positive elements along the (n − k)-band of S k (i.e., elements in positions (n − k + j, j), j = 1, . . . , k). It does this by successive row subtraction. It first eliminates the element in the (n − k + 1, 1) position by subtracting αk,n−k times the (n − k)th row from the (n − k + 1)st row. It then eliminates the element in the (n−k+2, 2) position by subtracting αk,n−k+1 times the (n−k+1)st row from the (n − k + 2)nd row. This does not effect the previous “elimination,” etc. Similarly, from L = C n−1 · · · C 1 as in Corollary 6.7, let T k = C n−1 · · · C k ,

k = 1, . . . , n − 1.

Thus L = T k C k−1 · · · C 1 and T k = T k+1 C k . The matrix T k is a full (n − k)-banded matrix. The difference here is that (C k )−1 uses column subtraction (rather than the row subtraction of

162


(Rk )−1 ) to take the (n − k)-banded matrix T k to the (n − k − 1)-banded matrix T k+1 . Starting from a unit diagonal lower strictly totally positive matrix it is readily verified that both the S k and T k are unit diagonal lower totally positive matrices and are as strictly totally positive as they can be, considering their geometric structure, i.e., considering the fact that they are (n − k)-banded. The matrix S k satisfies i1 , . . . , ir Sk >0 j1 , . . . , jr if and only if js ≤ is ≤ js + (n − k),

s = 1, . . . , r,

and the exact same inequalities hold for T k . Thus the S k and T k have exactly the same total positivity structure. We can state this procedure formally as follows. Proposition 6.8 Let P be an n × n unit diagonal lower triangular (n − k)banded totally positive matrix for which i1 , . . . , ir >0 P j1 , . . . , jr if and only if js ≤ is ≤ js + (n − k), s = 1, . . . , r. Then there exist Rk and C k where Rk = En−k+1,n−k (αn−k ) · · · En,n−1 (αn−1 ) with αj > 0, j = n − k, . . . , n − 1, and C k = E2,1 (β1 ) · · · Ek+1,k (βk ) with βj > 0, j = 1, . . . , k, such that P = Rk S = T C k where S and T are n × n unit diagonal lower triangular (n − k − 1)-banded totally positive matrices and i1 , . . . , ir i1 , . . . , ir , T >0 S j1 , . . . , jr j1 , . . . , jr if and only if js ≤ is ≤ js + (n − k − 1), s = 1, . . . , r.


163

We sequentially did the same thing at each step in Theorem 6.6 and 6.7, i.e., we decomposed S k = Rk S k+1 for k = 1, . . . , n − 1, in Theorem 6.6 and decomposed T k = T k+1 C k for k = 1, . . . , n − 1, in Corollary 6.7. However we could just as easily have decomposed S k via k S k = S k+1 C and T k via k T k+1 Tk = R k has the structure of C k , R k has the structure of Rk , and S k+1 , where C k+1 k+1 k+1 ,S and T all have the same structure. This is the content of T Proposition 6.8. Since we can make the choice of using row subtraction (Rk ) or column subtraction (C k ) at each step of the procedure we see that we actually have 2n−1 possible factorizations. (Note that there is an ordering to these 2n−1 factorizations. Consider R1 · · · Rn−1 C n−1 · · · C 1 and for each k delete one and only one of the Rk or C k . These are the possible forms of the factorizations where it is understood that in every distinct factorization the entries of the Rk or C k depend on the choice of the factorization.) What characterizes these 2n−1 factorizations is that the Rk and C k have exactly k positive off-diagonal entries, the last k entries or the first k entries, respectively, and this is minimal. On the other hand, without the minimality of nonzero terms there can be (and will be) a continuum of factorizations of a lower strictly totally positive n × n matrix into the product of (n − 1) 1-banded totally positive factors. For example, consider the matrix   1 0 0 A =  1 1 0 . 1 2 1 This matrix can be factored into a continuum of a product of two 1-banded unit diagonal lower totally positive matrices, namely for every x ∈ [1/2, 1]      1 0 0 1 0 0 1 0 0 A= 1 1 0 = 1−x 1 0  x 1 0 . 0 1/x 1 0 2 − 1/x 1 1 2 1

164


(Every 3 × 3 lower strictly totally positive matrix can be factored into a continuum of products of two 1-banded lower totally positive matrices.) What was done for a lower strictly totally positive matrix is also valid for an upper strictly totally positive matrix, with the obvious modifications. Thus, taking into consideration the above results together with the LDU factorization of a strictly totally positive matrix as detailed in Theorem 2.10, we can summarize as follows. Theorem 6.9 Let A be an n × n strictly totally positive matrix. Then A can be factored in many ways in the form A = L1 · · · Ln−1 DU 1 · · · U n−1 where D is a diagonal matrix whose diagonal entries are strictly positive, the Lk are 1-banded, unit diagonal, lower totally positive matrices, and the U k are 1-banded, unit diagonal, upper totally positive matrices. There are also many (in general 22n−2 ) different factorizations of this form where among the L1 , . . . , Ln−1 there is exactly one matrix with k strictly positive entries on its off-diagonal, k = 1, . . . , n − 1, and among the U 1 , . . . , U n−1 there is exactly one matrix with k strictly positive entries on its off-diagonal, k = 1, . . . , n − 1. We should also recall that every strictly totally positive (totally positive matrix) also has a factorization of the form A = U DL, where U is upper strictly totally positive (totally positive) and L is lower strictly totally positive (totally positive). Thus there are many, many factorizations of A as a product of 2n − 1 1-banded totally positive matrices.

6.3 Factorizations of totally positive matrices What happens when L is only a lower totally positive matrix? Do we still have the above factorizations? If L is nonsingular (let us assume that it is unit diagonal) then the answer is yes, although we may lose uniqueness of the Rk and C k . For example, if (L)n,1 = (L)n−1,1 = 0, then in the factorization L = R1 · · · Rn−1 not only is R1 = En,n−1 (α1,n−1 ) superfluous, but in fact it may be arbitrarily chosen with sufficiently small nonnegative values for α1,n−1 . (This choice will effect the entries of R2 , . . . , Rn−1 .) Nonetheless the proof

6.3 Factorizations of totally positive matrices

165

of Theorem 6.6 remains valid, modulo this type of proviso, because L is nonsingular and as we have seen in Proposition 1.15 and Theorem 1.19 if i + 1, . . . , i + r L =0 1, . . . , r then necessarily

L

j + 1, . . . , j + s 1, . . . , s

=0

for all j ≥ i and all s ≥ r, so that the equations in the proof of Theorem 6.6 are solvable. However, if L is singular then we cannot apply the previous reasoning. Nevertheless, from density, and paralleling the analysis of Theorem 2.12, we have the following. Theorem 6.10 An n×n lower triangular matrix L is lower totally positive if and only if it can be factored in the form L = R1 · · · Rn−1 where Rk is a 1-banded lower totally positive matrix with (Rk )i+1,i = 0, i = 1, . . . , n − k − 1. Similarly, L is lower totally positive if and only if it can be factored in the form L = C n−1 · · · C 1 where C k is a 1-banded lower totally positive matrix with (C k )i+1,i = 0, i = k + 1, . . . , n − 1. Proof If L has either of the above factorizations, then L is a lower totally positive matrix as it is a product of the lower totally positive matrices Rk (or C k ). It remains to prove the converse claims. A simple variation of Theorem 2.6 proves that there exist a sequence of lower strictly totally positive matrices (Lm ) which approximate L, i.e., such that lim Lm = L.

m→∞

From Theorem 6.6 each Lm can be factored in the form 1 n−1 Lm = Rm · · · Rm k where the Rm are nonsingular 1-banded lower totally positive matrices with

166


k (Rm )i+1,i = 0, i = 1, . . . , n − k − 1. By pre- and postmultiplying by positive diagonal matrices we can factor Lm in the form 1 ! mR !n−1 !m Lm = D ···R m

!k are stochastic (rows sums 1). As a consequence we also have where the R m ij (m) = d!ii (m) j

! m . Thus all entries in each for all i, where the d!ii (m) is the (i, i) entry of D of the factors are uniformly bounded. Now that we have bounds on all the entries of the matrices in the factorization we may take a subsequence along which each of the entries of these matrices converges. Denote their respective limits by D, R1 , . . . , Rn−1 and redefine R1 as DR1 . Then L = R1 · · · Rn−1 where Rk is a 1-banded lower totally positive matrix with (Rk )i+1,i = 0, i = 1, . . . , n − k − 1. Similarly it follows that L can be factored in the form L = C n−1 · · · C 1 where each C k is a 1-banded lower totally positive matrix that satisfies (C k )i+1,i = 0, i = k + 1, . . . , n − 1. As was seen in Section 6.2, these are just two of at least 2n−1 (generally distinct) possible factorizations. That is, we can mix and match the factors where for each k we choose to obtain a C k or an Rk . Note that if L is nonsingular then each of the Rk (or C k ) must be nonsingular. Thus if the L of Theorem 6.10 is unit diagonal, then we can also make this same demand on the Rk (or C k ). Paralleling Theorem 6.9, we can record the following. Theorem 6.11 Let A be an n × n totally positive matrix. Then A can be factored in many ways in the form A = L1 · · · Ln−1 U 1 · · · U n−1 where the Lk are 1-banded, unit diagonal, lower totally positive matrices, and the U k are 1-banded, unit diagonal, upper totally positive matrices. We also know that we can restrict the Lk and U k as in Theorem 6.10 so that they have a fairly large number of zero entries on their nonzero off-diagonal.

6.4 Remarks

167

6.4 Remarks The first factorization theorem for totally positive matrices is often called the Whitney Theorem. The name was given by Loewner [1955], although there is no evidence to indicate that Whitney [1951] ever thought of her result in this way. Whitney only talks about a “reduction theorem” and proves what we have listed as Proposition 1.9. Loewner gave this name to his factorization theorem because it is proved by repeated applications of Proposition 1.9. In fact the Whitney/Loewner method of factorization is not, from our perspective, a good factorization. Rather than eliminating successive off-diagonals, i.e., going from a (k + 1)-banded lower (upper) totally positive matrix to a k-banded lower (upper) totally positive matrix via multiplication by one 1-banded totally positive matrix, the procedure of Whitney/Loewner eliminates (makes zero) successive columns in a lower totally positive matrix. This implies the need for n(n − 1)/2 1-banded factors when factoring a lower totally positive matrix (rather than just the n − 1 factors). In the Whitney Theorem one assumes nonsingularity of the matrix. Cryer [1976] obtained 1-banded factorizations for an arbitrary totally positive matrix. But again the number of such factors can be very large. In fact Metelmann [1973] (the paper was almost unnoticed) was the first to state a factorization theorem for totally positive matrices with the correct number of factors. In Cavaretta, Dahmen, Micchelli, Smith [1981] can be found a factorization theorem (with the correct number of factors) for infinite strictly m-banded totally positive matrices. This was generalized in de Boor, Pinkus [1982] to an arbitrary totally positive nonsingular matrix (banded if the matrix is infinite). All three of these last mentioned papers, i.e., Metelmann [1973], Cavaretta, Dahmen, Micchelli, Smith [1981] and de Boor, Pinkus [1982] also consider totally positive matrices A = (aij ) that are (r, s)-banded, i.e., for which aij = 0 only if −s ≤ i − j ≤ r, and in this case show that one can make do with r + s factors (aside from the diagonal matrix). This also follows from the method of proof of the results in this chapter and Proposition 6.8. The proofs in some of the papers mentioned above tend to be rather different. The method of proof of Theorem 6.6 as presented here is basically to be found in Micchelli, Pinkus [1991]. In that paper a factorization is also given for n × m totally positive matrices. If n > m then aside from m − 1 lower and m − 1 upper triangular m × m 1-banded totally positive factors, there are also n − m totally positive factors that are k × (k − 1) matrices, k = m + 1, . . . , n, all of whose entries are zero except for those on the two main diagonals (i.e., the (i, i)- and (i + 1, i)-entries, i = 1, . . . , k − 1). This follows by expanding the

168


n × m totally positive matrix to an n × n totally positive matrix (just add columns of zeros), choosing the suitable factorization, and then truncating the result. In a series of papers (see Gasca, Pe˜ na [1996] and references therein), factorizations of totally positive matrices are studied in depth using Neville elimination. These authors also use factorizations to obtain many of the characterizations of totally positive matrices as presented in Chapter 2; see e.g., Gasca, Pe˜ na [1992], [1993]. In this monograph we have chosen a different approach. Factorizations are also used in the papers of Koev [2005], Demmel, Koev [2005] and Koev [2007] where their concern is with computing using totally positive matrices. For the use of factorizations in connection with weighted planar networks, see Fomin, Zelevinsky [1999] and Fomin, Zelevinsky [2000].

Afterword

It is very difficult, if not well nigh impossible, to give an exact history of the development of any set of ideas. Nonetheless, there are four persons whose contributions “stand out” when considering the history of total positivity. They are I. J. Schoenberg, M. G. Krein, F. R. Gantmacher, and S. Karlin. Of course they did not work in a vacuum and numerous influences are very evident in their research. It was Schoenberg who initiated the study of the variation diminishing properties of totally positive matrices in 1930 in Schoenberg [1930], and the study of P´ olya frequency functions in the late 1940s and early 1950s. Independently, and unaware of Schoenberg’s work, Krein was developing the theory of total positivity as it related to ordinary differential equations whose Green’s functions are totally positive. Furthermore, in the mid1930s Krein, together with Gantmacher, proved the spectral properties of totally positive kernels and matrices, and many other properties (see Gantmacher, Krein [1935], Gantmacher [1936], Gantmacher, Krein [1937], and their influential Gantmacher, Krein [1941], which was later reissued as Gantmacher, Krein [1950], and its translations in German in 1960 and in English in 1961 and 2002). These topics are the foundations upon which has been constructed the theory of total positivity. Karlin’s role was somewhat different. His books Karlin, Studden [1966] and Karlin [1968], the latter titled Total Positivity. Volume 1 (but there is no Volume 2), presented many new results and ideas and also synthesized and popularized many of these ideas. As the reader has hopefully noted, each chapter of this monograph ends with remarks that include bibliographical references and explanations. However I wanted to take this opportunity to write a “few words” in memory of each of these gentlemen. I. J. (Iso) Schoenberg (1903–1990) was born in Galatz, Romania and 169

170

Afterword

I. J. Schoenberg, 1903–1990

died in Madison, Wisconsin. His family moved to Jassy in 1910, and he studied mathematics at the university there. He spent three years in Germany studying with both Edmund Landau and Issai Schur. Influenced by Schur, he wrote a thesis in Analytic Number Theory. (From Landau he also took Landau’s daughter Charlotte (Dolli) as his first wife. He is one of many exemplars of the well-known fact that mathematical talent is an inherited trait: from father to son-in-law.) Schoenberg was interested in the problem of estimating the number of real zeros of a polynomial, and this led him to his work on variation diminishing transformations and P´ olya frequency functions and kernels, which are two major topics in the theory of total positivity. A Rockefeller fellowship fortunately brought him to the United States in 1930. He spent time at the University of Chicago, was a Fellow at the newly established Princeton Institute of Advanced Studies, was on the faculty of Colby College from 1936 to 1941, and was then at the University of Pennsylvania from 1941 to 1965. In 1965 Schoenberg moved to the University of Wisconsin and joined the Mathematics Research Center and the Department of Mathematics. He retired in 1973 but remained mathematically active until his death. He contributed to many areas of mathematics, in particular total positivity and splines. For further information, see de Boor [1988] (and especially Schoenberg’s autobiographical “A brief account of my life and work” at the beginning of the first volume), Askey, de Boor [1990], MacTutor, and references therein. Mark Grigorievich Krein (1907–1989) was born in Kiev and died in Odessa. He was a truly eminent and exceedingly prolific mathematician

Afterword

171

M. G. Krein, 1907–1989

who contributed significantly to and had a tremendous impact upon many different areas of mathematics (see e.g., Gohberg [1989] and Gohberg [1990], MacTutor, and references therein). The story of M. G. Krein, and the mathematical schools he built, is fundamentally marred by the tyranny and antisemitism to which he was constantly exposed. Krein was dismissed from his position at the University of Odessa in 1944 and from his parttime position at the Mathematical Institute of the Ukrainian Academy of Sciences in Kiev in 1952. We can only speculate on what might have been if he had been treated with the respect and dignity that were his due. In 1939 he was elected a corresponding member of the Ukrainian Academy of Sciences. He was never elected a full member. (This prompted the famous mathematical joke. Ques: How do you know that the Ukrainian Academy of Sciences is the best academy in the world? Ans: Because Krein is only a corresponding member.) From 1944 to 1954 Krein held the chair in theoretical mechanics at the Odessa Naval Engineering Institute, and from 1954 until his retirement he held the chair in theoretical mechanics at the Odessa Civil Engineering Institute. Despite being persecuted Krein received international recognition. He was elected an honorary member of the American Academy of Arts and Sciences in 1968, a foreign member of the National Academy of Sciences (of the USA) in 1979, and in 1982 he was awarded the Wolf Prize. The citation for this prize states: “Krein brought the full force of mathematical analysis to bear on problems of function theory, operator theory, probability and mathematical physics. His contributions led to important developments in the applications of mathematics to different fields ranging from theoretical mechanics to

172

Afterword

electrical engineering. His style in mathematics and his personal leadership and integrity have set standards of excellence.”

F. R. Gantmacher, 1908–1964

Feliks Ruvimovich Gantmacher (1908–1964) was born in Odessa and he studied there. In 1934 he moved to Moscow, where he resided until his death. Gantmacher is, of course, known for his excellent and influential book The Theory of Matrices (Gantmacher [1953]), and his book with Krein; Gantmacher, Krein [1941], [1950]. He was also one of the organizers and editors of the journal Uspekhi Mat. Nauk (Russian Math. Surveys). Gantmacher was instrumental in the establishment and organization of the well-known Moscow Physico-Technical Institute, where from 1953 until his death he headed the Department of Theoretical Applied Mathematics. An obituary on Gantmacher may be found in Gantmacher [1965].

S. Karlin, 1924–2007

Afterword

173

Samuel (Sam) Karlin (1924–2007) was born in Yonova, Poland, but he was raised in Chicago. He earned his PhD from Princeton in 1947 under the guidance of S. Bochner. From 1956 he was a faculty member at Stanford University. The breadth and depth of his interests and contributions in mathematics and in science are astounding. Karlin is the author of more than 450 papers and 10 books, and he had numerous doctoral students. Karlin was passionate about mathematics and science. His passion showed in his lectures, his lifestyle, and his interaction with students and colleagues. Karlin was widely honored, and he received honorary doctorates, numerous prizes (the John von Neumann Theory Prize in 1987, and the National Medal of Science in 1989 to name but two), and he was elected to various academies (see Karlin [2002] and MacTutor).

References

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Author index

Aissen, M., 125, 174 Ando, T., x, 33, 86, 152, 153, 174 Askey, R., 170, 174 Asner, B. A. Jr., 125, 174 Beckenbach, E. F., 34, 174 Bellman, R., 34, 174 Berenstein, A., ix, 174 Boocher, A., 34, 35, 174 de Boor, C., ix, 34, 167, 170, 174 Brenti, F., ix, 125, 174 Brown, L. D., 86, 174 Brualdi, R. A., 33, 175 Buslaev, A. P., 153, 175 Carlson, B. C., 125, 175 Carlson, D., 34, 175 Carnicer, J. M., 86, 175 Cavaretta, A. S. Jr., ix, 167, 175 Crans, A. S., 126, 175 Craven, T., 75, 175 Cryer, C., 74, 167, 175 Csordas, G., 75, 175 Dahmen, W. A., ix, 167, 175 Demmel, J., ix, 168, 175 Dimitrov, D. K., 75, 175

Garloff, J., 86, 126, 176 Gasca, M., 34, 74, 75, 168, 176–178 Gekhtman, M. I., 34, 35, 153, 175 Gladwell, G. M. L., 74, 75, 177 Gohberg, I., 171, 177 Goodman, T. N. T., ix, 86, 125, 175, 177 Gross, K. I., ix, 177 Gustafson, J. L., 125, 175 Holtz, O., 125, 177 Horn, R. A., 126, 177 Johnson, C. R., 34, 35, 126, 153, 175, 177 Johnstone, I. M., 86, 174 Karlin, S., ix, x, 33, 34, 74, 86, 125, 152, 153, 169, 173, 177, 178 Katkova, O. M., 75, 178 Kellogg, O. D., 33, 152, 178 Kemperman, J. H. B., 125, 178 Koev, P., ix, 168, 175, 178 Koteljanskii, D. M., 34, 153, 178 Krein, M. G., ix, x, 33, 34, 125, 152, 153, 169, 172, 176, 178 Kurtz, D. C., 126, 178

Edrei, A., 125, 175 Elias, U., 153, 175 Eveson, S. P., 153, 175

Loewner, K., 167, 178 Lusztig, G., ix, 178

Fallat, S. M., 34, 35, 126, 153, 175 Fan, K., 34, 86, 176 Fekete, M., 74, 176 Fomin, S., ix, 168, 174, 176 Friedland, S., 153, 176 Froehle, B., 34, 35, 174

M¨ uhlbach, G., 34, 178 MacGibbon, K. B., 86, 174 Mal´ o, E., 126, 178 Marden, M., 125, 178 Markham, T. L., 126, 178 Marshall, A. W., ix, 178 Metelmann, K., 74, 167, 178 Micchelli, C. A., ix, 34, 74, 167, 175–178 Motzkin, Th., 86, 178

Gantmacher, F. R., ix, x, 33, 34, 125, 152, 153, 169, 172, 176

180

Author index Nudel’man, A. A., x, 125, 178 Olkin, I., ix, 178 P´ olya, G., 74, 125, 176, 179 Pe˜ na, J. M., 34, 74, 75, 86, 168, 175, 177 Pinkus, A., ix, 34, 74, 153, 167, 174, 175, 177–179 Pitman, J., 125, 179 Rahman, Q. I., 125, 126, 179 Richards, D. St. P., ix, 177 Schmeisser, G., 125, 126, 179 Schneider, H., 33, 175 Schoenberg, I. J., x, 33, 85, 86, 125, 169, 174, 179 Schumaker, L. L., ix, 179 Shapiro, B. Z., 74, 179 Shapiro, M. Z., 74, 179

181

Shohat, J. A., 125, 179 Skandera, M., 34, 35, 179 Smith, P. W., ix, 167, 175, 179 Stieltjes, T. J., 102, 179 Studden, W. J., x, 125, 169, 178 Sun, Q., 125, 177 Szeg˝ o, G., 125, 179 Tamarkin, J. D., 125, 179 Vishnyakova, A. M., 75, 178 Wagner, D., 126, 176, 179 Wang, Y., 125, 179 Whitney, A. M., 34, 74, 86, 125, 167, 174, 179 Yeh, Y.-N., 125, 179 Zelevinsky, A., ix, 168, 174, 176

Subject index

LDU factorization, 50 almost strictly totally positive, 24, 34

lower strictly totally positive, 47 lower totally positive, 47 lower triangular, 47

banded, 155

Mal´ o’s Theorem, 123

Cauchy matrix, 92 Cauchy–Binet formula, 2, 33 Chebyshev system, 88 compound matrix, 2

oscillation matrix, 127

Descartes system, 88 dispersion, 37 eigenvalue interlacing, 140 exponentials, 88 Fekete’s Lemma, 37 Gantmacher–Krein Theorem, 130 Gaussian polynomials, 110 generalized Hadamard inequality, 24 generalized Hurwitz matrix, 111 Grassman product, 3 Green’s matrix, 96, 121 Hadamard inequality, 24, 34 Hadamard product, 119 Hankel matrix, 101, 123 Hurwitz matrix, 117, 124 Hurwitz polynomial, 117, 124

Perron’s Theorem, 130 pivot block, 4 principal minor, 5 principal submatrix, 4 Schur Product Theorem, 123 shadow, 13 sign changes, 76 sign regular, 86 strictly sign regular, 86 strictly totally positive, 2 strictly totally positive kernel, 87 Sylvester’s Determinant identity, 3 Szasz’s inequality, 34 Toeplitz matrix, 104, 123 totally positive, 2 totally positive kernel, 87 triangular total positivity, 47 upper strictly totally positive, 47 upper totally positive, 47 upper triangular, 47

Jacobi matrix, 97, 121

variation diminishing, 76, 85

Kronecker’s Theorem, 132

Whitney Theorem, 167

182

Totally Positive Matrices (Cambridge Tracts in Mathematics)

Totally Positive Matrices (Cambridge Tracts in Mathematics)

Totally positive matrices

Matrices of Sign-Solvable Linear Systems (Cambridge Tracts in Mathematics)

Divisors (Cambridge Tracts in Mathematics)

Convexity (Cambridge Tracts in Mathematics)

Divisors (Cambridge Tracts in Mathematics)

Totally nonnegative matrices

The Lebesgue Integral (Cambridge Tracts in Mathematics)

The Lévy Laplacian (Cambridge Tracts in Mathematics)

Hopf Algebras (Cambridge Tracts in Mathematics)

Simple Noetherian Rings (Cambridge Tracts in Mathematics)

Quasi-Frobenius Rings (Cambridge Tracts in Mathematics)

The Lebesgue Integral (Cambridge Tracts in Mathematics)

Totally Nonnegative Matrices (Princeton Series in Applied Mathematics)

Simple Noetherian Rings (Cambridge Tracts in Mathematics)

Proximity Spaces (Cambridge Tracts in Mathematics 59)

Forcing Idealized (Cambridge Tracts in Mathematics)

Multiple Forcing (Cambridge Tracts in Mathematics 88)

Sets of Multiples (Cambridge Tracts in Mathematics)

Forcing Idealized (Cambridge Tracts in Mathematics)

Asymptotic Expansions (Cambridge Tracts in Mathematics)

Fixed Point Theorems (Cambridge Tracts in Mathematics)

Positive Definite Matrices

Positive definite matrices

Completely positive matrices

Positive Definite Matrices

Completely positive matrices

Fourier Integrals in Classical Analysis (Cambridge Tracts in Mathematics)

Fourier Integrals in Classical Analysis (Cambridge Tracts in Mathematics)

Jordan Structures in Geometry and Analysis (Cambridge Tracts in Mathematics)

Totally Positive Matrices (Cambridge Tracts in Mathematics)